Circuit Construction¶
The Circuit
class forms the unit of computation that we can send off to a quantum coprocessor. Each instruction is to be performed in order, potentially parallelising when they use disjoint sets of (qu)bits. To capture this freedom of parallelisation, we treat the circuit as a Directed Acyclic Graph with a vertex for each instruction and directed edges following the paths of resources (e.g. qubits and bits) between them. This DAG representation describes the abstract circuit ignoring these trivial commutations/parallel instructions.
In general, we consider Circuit
instances to represent open circuits; that is, they can be used within arbitrary contexts, so any input state can be supplied and there is no assumption on how the output state should be used. In practice, when we send a Circuit
off to be executed, it will be run with all qubits in the initial state \(0\rangle^{\otimes n}\) and all bits set to \(0\), then the classical outputs returned and the quantum state discarded.
Each circuit can be represented as a POVM on the combined quantum/classical state space by composing the representations assigned to each basic instruction. However, many use cases will live predominantly in the pure quantum space where the operations are simply unitaries acting on the quantum state. One practical distinction between these cases is the relevance of global phase: something that cannot be identified at the POVM level but has importance for pure states as it affects how we interpret the system and has an observable difference when the system is then coherently controlled. For example, an Rz gate and a U1 gate give equivalent effects on the quantum state but have a different global phase, meaning their unitaries look different, and a controlledRz is different from a controlledU1. A Circuit
will track global phase to make working with pure quantum processes easier, though this becomes meaningless once measurements and other classical interaction are applied and has no impact on the instructions sent to a quantum device when we eventually run it.
Given the small scale and lack of dynamic quantum memories for both devices and simulations, we assume each qubit and bit is statically registered and hence each Circuit
has the same number of inputs as outputs. The set of data units (qubits and bits) used by the Circuit
is hence going to be constant, so we can define it upfront when we construct one. We can also optionally give it a name for easy identification.
from pytket import Circuit
trivial_circ = Circuit() # no qubits or bits
quantum_circ = Circuit(4) # 4 qubits and no bits
mixed_circ = Circuit(4, 2) # 4 qubits and 2 bits
named_circ = Circuit(2, 2, "my_circ")
Basic Gates¶
The bulk of the interaction with a Circuit
object will be in building up the sequence of instructions to be run. The simplest way to do this is by adding each instruction in execution order to the end of the circuit.
Basic quantum gates represent some unitary operation applied to some qubits. Adding them to a Circuit
just requires specifying which qubits you want to apply them to. For controlledgates, the convention is to give the control qubit(s) first, followed by the target qubit(s).
from pytket import Circuit
circ = Circuit(4) # qubits are numbered 03
circ.X(0) # first apply an X gate to qubit 0
circ.CX(1, 3) # and apply a CX gate with control qubit 1 and target qubit 3
circ.Z(3) # then apply a Z gate to qubit 3
circ.get_commands() # show the commands of the built circuit
[X q[0];, CX q[1], q[3];, Z q[3];]
For parameterised gates, such as rotations, the parameter is always given first. Because of the prevalence of rotations with angles given by fractions of \(\pi\) in practical quantum computing, the unit for all angular parameters is the halfturn (1 halfturn is equal to \(\pi\) radians).
from pytket import Circuit
circ = Circuit(2)
circ.Rx(0.5, 0) # Rx of angle pi/2 radians on qubit 0
circ.CRz(0.3, 1, 0) # ControlledRz of angle 0.3pi radians with
# control qubit 1 and target qubit 0
[Rx(0.5) q[0]; CRz(0.3) q[1], q[0]; ]
A large selection of common gates are available in this way, as listed in the API reference for the Circuit
class. However, for less commonly used gates, a wider variety is available using the OpType
enum, which can be added using the add_gate()
method.
from pytket import Circuit, OpType
circ = Circuit(5)
circ.add_gate(OpType.CnX, [0, 1, 4, 3])
# add controlledX with control qubits 0, 1, 4 and target qubit 3
circ.add_gate(OpType.XXPhase, 0.7, [0, 2])
# add e^{i (0.7 pi / 2) XX} on qubits 0 and 2
circ.add_gate(OpType.PhasedX, [0.1, 0.5], [3])
# adds Rz(0.5 pi); Rx(0.1 pi); Rz(0.5 pi) on qubit 3
[CnX q[0], q[1], q[4], q[3]; XXPhase(0.7) q[0], q[2]; PhasedX(3.9, 0.5) q[3]; ]
The API reference for the OpType
class details all available operations that can exist in a circuit.
In the above example, we asked for a PhasedX
with angles [0.1, 0.5]
, but received PhasedX(3.9, 0.5)
. pytket
will freely map angles into the range \(\left[0, r\right)\) for some range parameter \(r\) that depends on the OpType
, preserving the unitary matrix (including global phase).
Measurements¶
Measurements go a step further by interacting with both the quantum and classical data. The convention used in pytket
is that all measurements are nondestructive, singlequbit measurements in the \(Z\) basis; other forms of measurements can be constructed by combining these with other operations.
Adding a measurement works just like adding any other gate, where the first argument is the qubit to be measured and the second specifies the classical bit to store the result in.
from pytket import Circuit
circ = Circuit(4, 2)
circ.Measure(0, 0) # Zbasis measurement on qubit 0, saving result in bit 0
circ.CX(1, 2)
circ.CX(1, 3)
circ.H(1)
circ.Measure(1, 1) # Measurement of IXXX, saving result in bit 1
[Measure q[0] > c[0]; CX q[1], q[2]; CX q[1], q[3]; H q[1]; Measure q[1] > c[1]; ]
Because the classical bits are treated as statically assigned locations, writing to the same bit multiple times will overwrite the previous value.
from pytket import Circuit
circ = Circuit(2, 1)
circ.Measure(0, 0) # measure the first measurement
circ.CX(0, 1)
circ.Measure(1, 0) # overwrites the first result with a new measurement
[Measure q[0] > c[0]; CX q[0], q[1]; Measure q[1] > c[0]; ]
Depending on where we plan on running our circuits, the backend or simulator might have different requirements on the structure of measurements in the circuits. For example, statevector simulators will only work deterministically for purequantum circuits, so will fail if any measures are present at all. More crucially, nearterm quantum hardware almost always requires all measurements to occur in a single parallel layer at the end of the circuit (i.e. we cannot measure a qubit in the middle of the circuit).
from pytket import Circuit
circ0 = Circuit(2, 2) # all measurements at end
circ0.H(1)
circ0.Measure(0, 0)
circ0.Measure(1, 1)
circ1 = Circuit(2, 2) # this is DAGequivalent to circ1, so is still ok
circ1.Measure(0, 0)
circ1.H(1)
circ1.Measure(1, 1)
circ2 = Circuit(2, 2)
# reuses qubit 0 after measuring, so this may be rejected by a device
circ2.Measure(0, 0)
circ2.CX(0, 1)
circ2.Measure(1, 1)
circ3 = Circuit(2, 1)
# overwriting the classical value means we have to measure qubit 0
# before qubit 1; they won't occur simultaneously so this may be rejected
circ3.Measure(0, 0)
circ3.Measure(1, 0)
[Measure q[0] > c[0]; Measure q[1] > c[0]; ]
The simplest way to guarantee this is to finish the circuit by measuring all qubits. There is a shorthand function measure_all()
to make this easier.
from pytket import Circuit
# measure qubit 0 in Z basis and 1 in X basis
circ = Circuit(2, 2)
circ.H(1)
circ.measure_all()
# measure_all() adds bits if they are not already defined, so equivalently
circ = Circuit(2)
circ.H(1)
circ.measure_all()
[Measure q[0] > c[0]; H q[1]; Measure q[1] > c[1]; ]
On devices where midcircuit measurements are available, they may be highly noisy and not apply just a basic projector on the quantum state. We can view these as “effectively destructive” measurements, where the qubit still exists but is in a noisy state. In this case, it is recommended to actively reset a qubit after measurement if it is intended to be reused.
from pytket import Circuit, OpType
circ = Circuit(2, 2)
circ.Measure(0, 0)
# Actively reset state to 0>
circ.Reset(0)
# Conditionally flip state to 1> to reflect measurement result
circ.X(0, condition_bits=[0], condition_value=1)
# Use the qubit as if the measurement was nondestructive
circ.CX(0, 1)
[Measure q[0] > c[0]; Reset q[0]; IF ([c[0]] == 1) THEN X q[0]; CX q[0], q[1]; ]
Barriers¶
The concept of barriers comes from lowlevel classical programming. They exist as instructions but perform no active operation. Instead, their function is twofold:
At compiletime, prevent the compiler from reordering operations around the barrier.
At runtime, ensure that all operations before the barrier must have finished before any operations after the barrier start.
The intention is the same for Circuit
s. Inserting barriers can be used to segment the program to easily spot how it is modified during compilation, and some quantum hardware uses barriers as the primary method of embedding timing information.
Adding a barrier to a Circuit
is done using the add_barrier()
method. In general, a barrier is placed on some subset of the (qu)bits to impose these ordering restrictions on those (qu)bits specifically (i.e. we don’t care about reorders on the other (qu)bits).
from pytket import Circuit
circ = Circuit(4, 2)
circ.H(0)
circ.CX(1, 2)
circ.add_barrier([0, 1, 2, 3], [0, 1]) # add a barrier on all qubits and bits
circ.Measure(0, 0)
circ.Measure(2, 1)
[H q[0]; CX q[1], q[2]; Barrier q[0], q[1], q[2], q[3], c[0], c[1]; Measure q[0] > c[0]; Measure q[2] > c[1]; ]
Registers and IDs¶
Using integer values to refer to each of our qubits and bits works fine for smallscale experiments, but when building up larger and more complicated programs, it is much easier to manage if we are able to name the resources to attach semantic meaning to them and group them into related collections. pytket
enables this by supporting registers and named IDs.
Each unit resource is associated with a UnitID
(typically the subclasses Qubit
or Bit
), which gives a name and some (\(n\)dimensional) index. A (quantum/classical) register is hence some collection of UnitID
s with the same name, dimension of index, and type of associated resource. These identifiers are not necessarily tied to a specific Circuit
and can be reused between many of them.
Named resources can be added to Circuit
s individually, or by declaring a 1dimensional register. Any of the methods for adding gates can then use these IDs.
from pytket import Circuit, Qubit, Bit
circ = Circuit()
qreg = circ.add_q_register("reg", 2) # add a qubit register
anc = Qubit("ancilla") # add a named qubit
circ.add_qubit(anc)
par = Bit("parity", [0, 0]) # add a named bit with a 2D index
circ.add_bit(par)
circ.CX(qreg[0], anc) # add gates in terms of IDs
circ.CX(qreg[1], anc)
circ.Measure(anc, par)
[CX reg[0], ancilla; CX reg[1], ancilla; Measure ancilla > parity[0, 0]; ]
A Circuit
can be inspected to identify what qubits and bits it contains.
from pytket import Circuit, Qubit
circ = Circuit()
circ.add_q_register("a", 4)
circ.add_qubit(Qubit("b"))
circ.add_c_register("z", 3)
print(circ.qubits)
print(circ.bits)
[a[0], a[1], a[2], a[3], b]
[z[0], z[1], z[2]]
To help encourage consistency of identifiers, a Circuit
will reject a new (qu)bit or register if it disagrees with existing IDs with the same name; that is, it refers to a different resource type (qubit vs bit), the index has a different dimension, or some resource already exists with the exact same ID in the Circuit
. Identifiers with the same register name do not have to have contiguous indices (many devices require noncontiguous indices because qubits may be taken offline over the lifetime of the device).
from pytket import Circuit, Qubit, Bit
circ = Circuit()
# set up a circuit with qubit a[0]
circ.add_qubit(Qubit("a", 0))
# rejected because "a" is already a qubit register
circ.add_bit(Bit("a", 1))

RuntimeError Traceback (most recent call last)
Cell In[13], line 8
5 circ.add_qubit(Qubit("a", 0))
7 # rejected because "a" is already a qubit register
> 8 circ.add_bit(Bit("a", 1))
RuntimeError: Cannot add bit with ID "a[1]" as register is not compatible
# rejected because "a" is already a 1D register
circ.add_qubit(Qubit("a", [1, 2]))
circ.add_qubit(Qubit("a"))

RuntimeError Traceback (most recent call last)
Cell In[14], line 2
1 # rejected because "a" is already a 1D register
> 2 circ.add_qubit(Qubit("a", [1, 2]))
3 circ.add_qubit(Qubit("a"))
RuntimeError: Cannot add qubit with ID "a[1, 2]" as register is not compatible
# rejected because a[0] is already in the circuit
circ.add_qubit(Qubit("a", 0))

RuntimeError Traceback (most recent call last)
Cell In[15], line 2
1 # rejected because a[0] is already in the circuit
> 2 circ.add_qubit(Qubit("a", 0))
RuntimeError: A unit with ID "a[0]" already exists
The basic integer identifiers are actually a special case, referring to the default qubit (q[i]
) and bit (c[i]
) registers. We can create the UnitID
using the nameless Qubit
and Bit
constructors.
from pytket import Circuit, Qubit, Bit
circ = Circuit(4, 2)
circ.CX(Qubit(0), Qubit("q", 1)) # same as circ.CX(0, 1)
circ.Measure(Qubit(2), Bit("c", 0)) # same as circ.Measure(2, 0)
[Measure q[2] > c[0]; CX q[0], q[1]; ]
In some circumstances, it may be useful to rename the resources in the Circuit
. Given a partial map on UnitID
s, rename_units()
will change the association of IDs to resources (as long as the final labelling would still have consistent types for all registers). Any unspecified IDs will be preserved.
from pytket import Circuit, Qubit, Bit
circ = Circuit(2, 2)
circ.add_qubit(Qubit("a", 0))
qubit_map = {
Qubit("a", 0) : Qubit(3),
Qubit(1) : Qubit("a", 0),
Bit(0) : Bit("z", [0, 1]),
}
circ.rename_units(qubit_map)
print(circ.qubits)
print(circ.bits)
[a[0], q[0], q[3]]
[c[1], z[0, 1]]
Composing Circuits¶
Because Circuit
s are defined to have open inputs and outputs, it is perfectly natural to compose them by unifying the outputs of one with the inputs of another. Appending one Circuit
to the end of another matches the inputs and outputs with the same UnitID
.
from pytket import Circuit, Qubit, Bit
circ = Circuit(2, 2)
circ.CX(0, 1)
circ.Rz(0.3, 1)
circ.CX(0, 1)
measures = Circuit(2, 2)
measures.H(1)
measures.measure_all()
circ.append(measures)
circ
[CX q[0], q[1]; Rz(0.3) q[1]; CX q[0], q[1]; Measure q[0] > c[0]; H q[1]; Measure q[1] > c[1]; ]
Note
If one Circuit
lacks some unit present in the other, then we treat it as if it is an identity on that unit. In the extreme case where the Circuit
s are defined with disjoint sets of UnitID
s, the Circuit.append()
method will compose them in parallel.
To compose two circuits in parallel we can take tensor product using the * operator. This requires that the qubits in the circuits have distinct names.
from pytket import Circuit
from pytket.circuit.display import render_circuit_jupyter as draw
circ1 = Circuit()
j = circ1.add_q_register("j", 1)
circ1.Y(j[0])
circ2 = Circuit()
k = circ2.add_q_register("k", 2)
circ2.X(k[1])
circ2.CRz(0.64, k[1], k[0])
circ3 = circ1 * circ2 # Take the tensor product
draw(circ3)
If we attempt to form the tensor product of two circuits without distinct qubit names then we will get a RuntimeError
as the composition is not defined.
from pytket import Circuit
circ_x = Circuit()
l_reg1 = circ_x.add_q_register("l", 1)
circ_y = Circuit()
l_reg2 = circ_y.add_q_register("l", 1)
circ_x * circ_y # Error as both circuits have l[0]

RuntimeError Traceback (most recent call last)
Cell In[20], line 9
6 circ_y = Circuit()
7 l_reg2 = circ_y.add_q_register("l", 1)
> 9 circ_x * circ_y # Error as both circuits have l[0]
RuntimeError: Cannot merge circuits as both contain unit: l[0]
To change which units get unified, we could use Circuit.rename_units()
as seen before. In the case where we just want to append a subcircuit like a gate, we can do this with Circuit.add_circuit()
.
from pytket import Circuit, Qubit
circ = Circuit()
a = circ.add_q_register("a", 2)
circ.Rx(0.2, a[0])
circ.CX(a[0], a[1])
next_circ = Circuit(2)
next_circ.Z(0)
next_circ.CZ(1, 0)
circ.add_circuit(next_circ, [a[1], a[0]])
# This is equivalent to:
# temp = next_circ.copy()
# temp.rename_units({Qubit(0) : a[1], Qubit(1) : a[0]})
# circ.append(temp)
draw(circ)
Note
This requires the subcircuit to be defined only over the default registers so that the list of arguments given to Circuit.add_circuit()
can easily be mapped.
Statevectors and Unitaries¶
When working with quantum circuits we may want access to the quantum state prepared by our circuit. This can be helpful if we want to check whether our circuit construction is correct. The Circuit.get_statevector()
method will produce the statevector of our system after the circuit is applied. Here it is assumed that all the qubits are initialised in the \(0\rangle^{\otimes n}\) state.
from pytket import Circuit
circ = Circuit(2)
circ.H(0).CX(0, 1)
circ.get_statevector()
array([0.70710678+0.j, 0. +0.j, 0. +0.j, 0.70710678+0.j])
In addition Circuit.get_unitary()
can be used to numerically calculate the unitary matrix that will be applied by the circuit.
from pytket import Circuit
circ = Circuit(2)
circ.H(0).CZ(0, 1).H(1)
circ.get_unitary()
array([[ 0.5+0.j, 0.5+0.j, 0.5+0.j, 0.5+0.j],
[ 0.5+0.j, 0.5+0.j, 0.5+0.j, 0.5+0.j],
[ 0.5+0.j, 0.5+0.j, 0.5+0.j, 0.5+0.j],
[ 0.5+0.j, 0.5+0.j, 0.5+0.j, 0.5+0.j]])
Warning
The unitary matrix of a quantum circuit is of dimension \((2^n \times 2^n)\) where \(n\) is the number of qubits. The statevector will be a column vector with \(2^n\) entries . Due to this exponential scaling it will in general be very inefficient to compute the unitary (or statevector) of a circuit. These functions are intended to be used for sanity checks and spotting mistakes in small circuits.
Analysing Circuits¶
After creating a Circuit
, we will typically want to inspect what we have constructed to ensure that it agrees with the design we planned. The most basic form of this is to just get the object to return the sequence of operations back to us. Iterating through the Circuit
object will give back the operations as Command
s (specifying the operations performed and what (qu)bits they are performed on).
Because the Circuit
class identifies circuits up to DAG equivalence, the sequence will be some topological sort of the DAG, but not necessarily identical to the order the operations were added to the Circuit
.
from pytket import Circuit
circ = Circuit(3)
circ.CX(0, 1).CZ(1, 2).X(1).Rx(0.3, 0)
for com in circ: # equivalently, circ.get_commands()
print(com.op, com.op.type, com.args)
# NOTE: com is not a reference to something inside circ; this cannot be used to modify the circuit
CX OpType.CX [q[0], q[1]]
Rx(0.3) OpType.Rx [q[0]]
CZ OpType.CZ [q[1], q[2]]
X OpType.X [q[1]]
If you are working in a Jupyter environment, a Circuit
can be rendered using html for inline display.
from pytket import Circuit
from pytket.circuit.display import render_circuit_jupyter as draw
circ = Circuit(3)
circ.CX(0, 1).CZ(1, 2).X(1).Rx(0.3, 0)
draw(circ) # Render interactive circuit diagram
Note
The pytket circuit renderer can represent circuits in the standard circuit model or in the ZX representation. Other interactive features include adjustable zoom, circuit wrapping and image export.
pytket
also features methods to visualise the underlying circuit DAG graphically for easier visual inspection.
from pytket import Circuit
from pytket.utils import Graph
circ = Circuit(3)
circ.CX(0, 1).CZ(1, 2).X(1).Rx(0.3, 0)
Graph(circ).get_DAG() # Displays in interactive python notebooks
The visualisation tool can also describe the interaction graph of a Circuit
consisting of only one and twoqubit gates – that is, the graph of which qubits will share a twoqubit gate at some point during execution.
Note
The visualisations above are shown in ipython notebook cells. When working with a normal python script one can view rendered circuits in the browser with the view_browser()
function from the display module.
There are also the methods save_DAG()
and view_DAG()
for saving and visualising the circuit DAG.
from pytket import Circuit
from pytket.utils import Graph
circ = Circuit(4)
circ.CX(0, 1).CZ(1, 2).ZZPhase(0.63, 2, 3).CX(1, 3).CY(0, 1)
Graph(circ).get_qubit_graph()
The full instruction sequence may often be too much detail for a lot of needs, especially for large circuits. Common circuit metrics like gate count and depth are used to approximate the difficulty of running it on a device, providing some basic tools to help distinguish different implementations of a given algorithm.
from pytket import Circuit
circ = Circuit(3)
circ.CX(0, 1).CZ(1, 2).X(1).Rx(0.3, 0)
print("Total gate count =", circ.n_gates)
print("Circuit depth =", circ.depth())
Total gate count = 4
Circuit depth = 3
As characteristics of a Circuit
go, these are pretty basic. In terms of approximating the noise level, they fail heavily from weighting all gates evenly when, in fact, some will be much harder to implement than others. For example, in the NISQ era, we find that most technologies provide good singlequbit gate times and fidelities, with twoqubit gates being much slower and noisier [^cite_arut2019]. On the other hand, looking forward to the faulttolerant regime we will expect Clifford gates to be very cheap but the magic \(T\) gates to require expensive distillation procedures [1] [2].
We can use the OpType
enum class to look for the number of gates of a particular type. Additionally, the methods n_1qb_gates()
, n_2qb_gates()
and n_nqb_gates()
can be used to count the number of gates in terms of how many qubits they act upon irrespective of type.
We also define \(G\)depth (for a subset of gate types \(G\)) as the minimum number of layers of gates in \(G\) required to run the Circuit
, allowing for topological reorderings. Specific cases of this like \(T\)depth and \(CX\)depth are common to the literature on circuit simplification [3] [4].
from pytket import Circuit, OpType
from pytket.circuit.display import render_circuit_jupyter as draw
circ = Circuit(3)
circ.T(0)
circ.CX(0, 1)
circ.CX(2, 0)
circ.add_gate(OpType.CnRy, [0.6], [0, 1, 2])
circ.T(2)
circ.CZ(0, 1)
circ.CZ(1, 2)
circ.T(1)
draw(circ) # draw circuit diagram
print("T gate count =", circ.n_gates_of_type(OpType.T))
print("#1qb gates =", circ.n_1qb_gates())
print("#2qb gates =", circ.n_2qb_gates())
print("#3qb gates =", circ.n_nqb_gates(3)) # count the single CnRy gate (n=3)
print("T gate depth =", circ.depth_by_type(OpType.T))
print("2qb gate depth =", circ.depth_by_type({OpType.CX, OpType.CZ}))
T gate count = 3
#1qb gates = 3
#2qb gates = 4
#3qb gates = 1
T gate depth = 3
2qb gate depth = 4
Note
Each of these metrics will analyse the Circuit
“as is”, so they will consider each Box as a single unit rather than breaking it down into basic gates, nor will they perform any nontrivial gate commutations (those that don’t just follow by deformation of the DAG) or gate decompositions (e.g. recognising that a \(CZ\) gate would contribute 1 to \(CX\)count in practice).
Its also possible to count all the occurrences of each OpType
using the gate_counts()
function from the pytket.utils
module.
from pytket.utils.stats import gate_counts
gate_counts(circ)
Counter({<OpType.T: 27>: 3,
<OpType.CX: 45>: 2,
<OpType.CZ: 47>: 2,
<OpType.CnRy: 83>: 1})
We obtain a collections.Counter
object where the keys are the various OpType
s and the values represent how frequently each OpType
appears in our Circuit
. This method summarises the gate counts obtained for the circuit shown above.
Boxes¶
Working with individual basic gates is sufficient for implementing arbitrary circuits, but that doesn’t mean it is the most convenient option. It is generally far easier to argue the correctness of a circuit’s design when it is constructed using higherlevel constructions. In pytket
, the concept of a “Box” is to abstract away such complex structures as blackboxes within larger circuits.
Defining higher level subroutines as boxes is also beneficial from a circuit optimisation point of view. If the compiler can identify higher level structure in the circuit, this can be exploited to reduce the number of elementary gates in the compiled circuit. Examples of such optimisations can be seen in ToffoliBox
which permutes the computational basis states and ConjugationBox
which allows for more efficient controlled gates by exploiting circuit symmetry.
Circuit Boxes¶
The simplest example of this is a CircBox
, which wraps up another Circuit
defined elsewhere into a single blackbox. The difference between adding a CircBox
and just appending the Circuit
is that the CircBox
allows us to wrap up and abstract away the internal structure of the subcircuit we are adding so it appears as if it were a single gate when we view the main Circuit
.
Let’s first build a basic quantum circuit which implements a simplified version of a Grover oracle and then add it to another circuit as part of a larger algorithm.
from pytket.circuit import Circuit, OpType
from pytket.circuit.display import render_circuit_jupyter as draw
oracle_circ = Circuit(3, name="Oracle")
oracle_circ.X(0)
oracle_circ.X(1)
oracle_circ.X(2)
oracle_circ.add_gate(OpType.CnZ, [0, 1, 2])
oracle_circ.X(0)
oracle_circ.X(1)
oracle_circ.X(2)
draw(oracle_circ)
Now that we’ve built our circuit we can wrap it up in a CircBox
and add it to a another circuit as a subroutine.
from pytket.circuit import CircBox
oracle_box = CircBox(oracle_circ)
circ = Circuit(3)
circ.H(0).H(1).H(2)
circ.add_gate(oracle_box, [0, 1, 2])
draw(circ)
See how the name of the circuit appears in the rendered circuit diagram. Clicking on the box will show the underlying circuit.
Note
Despite the Circuit
class having methods for adding each type of box, the Circuit.add_gate()
is sufficiently general to append any pytket OpType to a Circuit
.
When constructing subroutines to implement quantum algorithms it is natural to distinguish different groups of qubits. For instance, in the quantum phase estimation algorithm (QPE) we would want to distinguish between state preparation qubits and ancillary qubits which are measured to yield an approximation of the phase. The QPE can then be used as a subroutine in other algorithms: for example, integer factoring or estimating the ground state energy of some molecule. For more on the phase estimation algorithm see the QPE example notebook.
For such algorithms we may wish to create a CircBox
containing qubit registers with distinct names. Below we will show construction of a simplified quantum phase estimation circuit which we will then turn into a subroutine.
from pytket.circuit import Circuit
# Set up circuit registers
qpe_circ = Circuit(name="QPE")
a = qpe_circ.add_q_register("a", 2)
s = qpe_circ.add_q_register("s", 1)
c = qpe_circ.add_c_register("c", 2)
# Initial superposition
qpe_circ.H(a[0])
qpe_circ.H(a[1])
# Sequence of controlled unitaries
qpe_circ.CU1(0.94, a[1], s[0])
qpe_circ.CU1(0.94, a[0], s[0])
qpe_circ.CU1(0.94, a[0], s[0])
# 2qubit QFT (simplified)
qpe_circ.H(a[0])
qpe_circ.CU1(0.5, a[1], a[0])
qpe_circ.H(a[1])
qpe_circ.SWAP(a[0], a[1])
# Measure qubits writing to the classical register
qpe_circ.measure_register(a, "c")
draw(qpe_circ)
Now that we have defined our phase estimation circuit we can use a CircBox
to define a reusable subroutine. This CircBox
will contain the state preparation and ancilla registers.
from pytket.circuit import CircBox
# Construct QPE subroutine
qpe_box = CircBox(qpe_circ)
Let’s now create a circuit to implement the QPE algorithm where we prepare the \(1\rangle\) state in the state prep register with a single X gate.
# Construct simplified state preparation circuit
algorithm_circ = Circuit()
ancillas = algorithm_circ.add_q_register("ancillas", 2)
state = algorithm_circ.add_q_register("state", 1)
c = algorithm_circ.add_c_register("c", 2)
algorithm_circ.X(state[0])
draw(algorithm_circ)
We can then compose our subroutine registerwise by using Circuit.add_circbox_with_regmap()
method.
To use the method, we pass in a python dictionary which maps the registers inside the box to those outside. The keys are the register names inside the CircBox
and the values are the register names of the external Circuit
.
Note that the sizes of the registers used as keys and values must be equal.
# Append QPE subroutine to algorithm_circ registerwise
algorithm_circ.add_circbox_with_regmap(
qpe_box, qregmap={"a": "ancillas", "s": "state"}, cregmap={"c": "c"}
)
draw(algorithm_circ)
Click on the QPE box in the diagram above to view the underlying circuit.
If we want add a CircBox
across multiple registers we can do this with the Circuit.add_circbox_regwise()
method.
Lets first define a circuit with the register names a
, b
and c
.
# Set up Circuit with registers a_reg, b_reg and c_reg
abc_circuit = Circuit()
a_reg = abc_circuit.add_q_register("a", 2)
b_reg = abc_circuit.add_q_register("b", 2)
c_reg = abc_circuit.add_q_register("c", 2)
# Add some gates
abc_circuit.H(a_reg[0])
abc_circuit.Ry(0.46, a_reg[1])
abc_circuit.CCX(a_reg[0], a_reg[1], c_reg[0])
draw(abc_circuit)
Now lets create a CircBox
containing some elementary gates and append it across the b
and c
registers with Circuit.add_circbox_regwise()
.
# Create subroutine
sub_circuit = Circuit(4, name="BC")
sub_circuit.CX(3, 2).CX(3, 1).CX(3, 0)
sub_circuit.H(3)
bc_subroutine = CircBox(sub_circuit)
# Append CircBox to the b_reg and c_reg registers (note empty list for classical registers)
abc_circuit.add_circbox_regwise(bc_subroutine, [b_reg, c_reg], [])
draw(abc_circuit)
Boxes for Unitary Synthesis¶
It is possible to specify small unitaries from numpy
arrays and embed them directly into circuits as boxes, which can then be synthesised into gate sequences during compilation.
from pytket.circuit import Circuit, Unitary1qBox, Unitary2qBox
import numpy as np
u1 = np.asarray([[2/3, (2+1j)/3],
[(2+1j)/3, 2/3]])
u1box = Unitary1qBox(u1)
u2 = np.asarray([[0, 1, 0, 0],
[0, 0, 0, 1],
[1, 0, 0, 0],
[0, 0, 1j, 0]])
u2box = Unitary2qBox(u2)
circ = Circuit(3)
circ.add_unitary1qbox(u1box, 0)
circ.add_unitary2qbox(u2box, 1, 2)
circ.add_unitary1qbox(u1box, 2)
circ.add_unitary2qbox(u2box, 1, 0)
draw(circ)
Note
For performance reasons pytket currently only supports unitary synthesis up to three qubits. Threequbit synthesis can be accomplished with Unitary3qBox
using a similar syntax.
Also in this category of synthesis boxes is DiagonalBox
. This allows synthesis of circuits for diagonal unitaries.
This box can be constructed by passing in a \((1 \times 2^n)\) numpy array representing the diagonal entries of the desired unitary matrix.
Controlled Box Operations¶
If our subcircuit is a pure quantum circuit (i.e. it corresponds to a unitary operation), we can construct the controlled version that is applied coherently according to some set of control qubits. If all control qubits are in the \(1\rangle\) state, then the unitary is applied to the target system, otherwise it acts as an identity.
from pytket.circuit import Circuit, CircBox, QControlBox
sub = Circuit(2, name="V")
sub.CX(0, 1).Rz(0.2, 1).CX(0, 1)
sub_box = CircBox(sub)
# Define the controlled operation with 2 control qubits
cont = QControlBox(sub_box, 2)
circ = Circuit(4)
circ.add_gate(sub_box, [2, 3])
circ.Ry(0.3, 0).Ry(0.8, 1)
# Add to circuit with controls q[0], q[1], and targets q[2], q[3]
circ.add_gate(cont, [0, 1, 2, 3])
draw(circ)
As well as creating controlled boxes, we can create a controlled version of an arbitrary Op
as follows.
from pytket.circuit import Op, OpType, QControlBox
op = Op.create(OpType.S)
ccs = QControlBox(op, 2)
Note
Whilst adding a control qubit is asymptotically efficient, the gate overhead is significant and can be hard to synthesise optimally, so using these constructions in a NISQ context should be done with caution.
In addition, we can construct a QControlBox
from any other pure quantum box type in pytket.
For example, we can construct a multicontrolled \(\sqrt{Y}\) operation as by first synthesising the base unitary with Unitary1qBox
and then constructing a QControlBox
from the box implementing \(\sqrt{Y}\).
from pytket.circuit import Unitary1qBox, QControlBox
import numpy as np
# Unitary for sqrt(Y)
sqrt_y = np.asarray([[1/2+1j/2, 1/21j/2],
[1/2+1j/2, 1/2+1j/2]])
sqrt_y_box = Unitary1qBox(sqrt_y)
c2_root_y = QControlBox(sqrt_y_box, 2)
Normally when we deal with controlled gates we implicitly assume that the control state is the “all \(1\rangle\)” state. So that the base gate is applied when all of the control qubits are all set to \(1\rangle\).
However its often useful to the flexibility to define the control state as some string of zeros and ones. Certain approaches to quantum algorithms with linear combination of unitaries (LCU) frequently make use of such gates.
A QControlBox
accepts an optional control_state
argument in the constructor. This is either a list of binary values or a single (bigendian) integer representing the binary string.
Lets now construct a multicontrolled Rz gate with the control state \(0010\rangle\). This means that the base operation will only be applied if the control qubits are in the state \(0010\rangle\).
from pytket.circuit import Circuit, Op, OpType, QControlBox
rz_op = Op.create(OpType.Rz, 0.61)
multi_controlled_rz = QControlBox(rz_op, n_controls=4, control_state=[0, 0, 1, 0])
test_circ = Circuit(5)
test_circ.add_gate(multi_controlled_rz, test_circ.qubits)
draw(test_circ)
Notice how the circuit renderer shows both filled and unfilled circles on the control qubits. Filled circles correspond to \(1\rangle\) controls whereas empty circles represent \(0\rangle\). As pytket uses the bigendian ordering convention we read off the control state from the top to the bottom of the circuit.
Pauli Exponential Boxes¶
Another notable construct that is common to many algorithms and highlevel circuit descriptions is the exponential of a Pauli tensor:
These occur very naturally in Trotterising evolution operators and native device operations.
from pytket.circuit import PauliExpBox
from pytket.pauli import Pauli
# Construct a PauliExpBox with a list of Paulis followed by the phase theta
xyyz = PauliExpBox([Pauli.X, Pauli.Y, Pauli.Y, Pauli.Z], 0.2)
pauli_circ = Circuit(4)
pauli_circ.add_gate(xyyz, [0, 1, 2, 3])
draw(pauli_circ)
To understand what happens inside a PauliExpBox
let’s take a look at the underlying circuit for \(e^{i \frac{\pi}{2}\theta XYYZ}\)
from pytket.passes import DecomposeBoxes
DecomposeBoxes().apply(pauli_circ)
draw(pauli_circ)
All Pauli exponentials of the form above can be implemented in terms of a single Rz(\(\theta\)) rotation and a symmetric chain of CX gates on either side together with some single qubit basis rotations. This class of circuit is called a Pauli gadget. The subset of these circuits corresponding to “Z only” Pauli strings are referred to as phase gadgets.
We see that the Pauli exponential \(e^{i\frac{\pi}{2} \theta \text{XYYZ}}\) has basis rotations on the first three qubits. The V and Vdg gates rotate from the default Z basis to the Y basis and the Hadamard gate serves to change to the X basis.
These Pauli gadget circuits have interesting algebraic properties which are useful for circuit optimisation.
A Pauli gadget can be expressed as \(V \, A \, V^\dagger\) where \(V\) is the the circuit composed of CX gates and single qubit basis rotations on the right hand side of the Rz gate and \(A\) is the Rz gate itself. This observation allows one to construct controlled Pauli gadgets much more efficently. See the blog post on the ConjugationBox
construct for more details.
For further discussion see the research publication on phase gadget synthesis [5]. Ideas from this paper are implemented in TKET as the OptimisePhaseGadgets and PauliSimp optimisation passes.
Phase Polynomials¶
Now we move on to discuss another class of quantum circuits known as phase polynomials. Phase polynomial circuits are a special type of circuits that use the {CX, Rz} gateset.
A phase polynomial \(p(x)\) is defined as a weighted sum of Boolean linear functions \(f_i(x)\):
A phase polynomial circuit \(C\) has the following action on computational basis states \(x\rangle\):
Such a phase polynomial circuit can be synthesisied in pytket using the PhasePolyBox
. A PhasePolyBox
is constructed using the number of qubits, the qubit indices and a dictionary indicating whether or not a phase should be applied to specific qubits.
Finally a linear_transfromation
parameter needs to be specified: this is a matrix encoding the linear permutation between the bitstrings \(x\rangle\) and \(g(x)\rangle\) in the equation above.
from pytket.circuit import PhasePolyBox
phase_poly_circ = Circuit(3)
qubit_indices = {Qubit(0): 0, Qubit(1): 1, Qubit(2): 2}
phase_polynomial = {
(True, False, True): 0.333,
(False, False, True): 0.05,
(False, True, False): 1.05,
}
n_qb = 3
linear_transformation = np.array([[1, 1, 0], [0, 1, 0], [0, 0, 1]])
p_box = PhasePolyBox(n_qb, qubit_indices, phase_polynomial, linear_transformation)
phase_poly_circ.add_gate(p_box, [0, 1, 2])
draw(p_box.get_circuit())
Multiplexors, Arbitrary State Preparation and ToffoliBox
¶
In the context of quantum circuits a multiplexor is type of generalised multicontrolled gate. Multiplexors grant us the flexibility to specify different operations on target qubits for different control states. To create a multiplexor we simply construct a dictionary where the keys are the state of the control qubits and the values represent the operation performed on the target.
Lets implement a multiplexor with the following logic. Here we treat the first two qubits as controls and the third qubit as the target.
if control qubits in \(00\rangle\):
: do Rz(0.3) on the third qubit
else if control qubits in \(11\rangle\):
: do H on the third qubit
else:
: do identity (i.e. do nothing)
from pytket.circuit import Op, OpType, MultiplexorBox
# Define both gates as an Op
rz_op = Op.create(OpType.Rz, 0.3)
h_op = Op.create(OpType.H)
op_map = {(0, 0): rz_op, (1, 1): h_op}
multiplexor = MultiplexorBox(op_map)
multi_circ = Circuit(3)
multi_circ.X(0).X(1) # Put both control qubits in the state 1>
multi_circ.add_gate(multiplexor, [0, 1, 2])
draw(multi_circ)
Notice how in the example above the control qubits are both in the \(1\rangle\) state and so the multiplexor applies the Hadamard operation to the third qubit. If we calculate our statevector we see that the third qubit is in the \(+\rangle = H0\rangle\) state.
# Assume all qubits initialised to 0> here
# Amplitudes of +> approx 0.707...
print("Statevector =", np.round(multi_circ.get_statevector().real, 4))
Statevector = [0. 0. 0. 0. 0. 0. 0.7071 0.7071]
In addition to the general MultiplexorBox
pytket has several other type of multiplexor box operations available.
Multiplexor 
Description 

The most general type of multiplexor (see above). 

Multiplexor where the operation applied to the target is a rotation gate about a single axis. 

Multiplexor for uniformly controlled single qubit gates ( \(U(2)\) operations). 

Multiplexor where the operation applied to the target is a tensor product of single qubit gates. 
One place where multiplexor operations are useful is in state preparation algorithms.
TKET supports the preparation of arbitrary quantum states via the StatePreparationBox
. This box takes a \((1\times 2^n)\) numpy array representing the \(n\) qubit statevector where the entries represent the amplitudes of the quantum state.
Given the vector of amplitudes TKET will construct a box containing a sequence of multiplexors using the method outlined in [6].
To demonstrate StatePreparationBox
let’s use it to prepare the W state.
from pytket.circuit import StatePreparationBox
w_state = 1 / np.sqrt(3) * np.array([0, 1, 1, 0, 1, 0, 0, 0])
w_state_box = StatePreparationBox(w_state)
state_circ = Circuit(3)
state_circ.add_gate(w_state_box, [0, 1, 2])
[StatePreparationBox q[0], q[1], q[2]; ]
# Verify state preperation
np.round(state_circ.get_statevector().real, 3) # 1/sqrt(3) approx 0.577
array([0. , 0.577, 0.577, 0. , 0.577, 0. , 0. , 0. ])
Note
Generic state preperation circuits can be very complex with the gatecount and depth increasing rapidly with the size of the state. In the special case where the desired state has only realvalued amplitudes, only multiplexed Ry operations are needed to accomplish the state preparation.
For some use cases it may be desirable to reset all qubits to the \(0\rangle\) state prior to state preparation. This can be done using the with_initial_reset
flag.
# Ensure all qubits initialised to 0>
w_state_box_reset = StatePreparationBox(w_state, with_initial_reset=True)
Finally let’s consider another box type, namely the ToffoliBox
. This box can be used to prepare an arbitrary permutation of the computational basis states.
To construct the box we need to specify the permutation as a keyvalue pair where the key is the input basis state and the value is output.
Let’s construct a ToffoliBox
to perform the following mapping:
We can construct a ToffoliBox
with a python dictionary where the basis states above are entered as keyvalue pairs.
For correctness if a basis state appears as key in the permutation dictionary then it must also appear and a value.
from pytket.circuit import ToffoliBox
# Specify the desired permutation of the basis states
mapping = {
(0, 0, 1): (1, 1, 1),
(1, 1, 1): (0, 0, 1),
(1, 0, 0): (0, 0, 0),
(0, 0, 0): (1, 0, 0),
}
# Define box to perform the permutation
perm_box = ToffoliBox(permutation=mapping)
This permutation of basis states can be achieved with purely classical operations {X, CCX}, hence the name ToffoliBox
.
In pytket however, the permutation is implemented efficently using a sequence of multiplexed rotations followed by a DiagonalBox
.
draw(perm_box.get_circuit())
Finally let’s append the ToffoliBox
onto our circuit preparing our w state to perform the permutation of basis states specified above.
state_circ.add_gate(perm_box, [0, 1, 2])
draw(state_circ)
np.round(state_circ.get_statevector().real, 3)
array([ 0.577, 0. , 0.577, 0. , 0. , 0. , 0. , 0.577])
Looking at the statevector calculation we see that our ToffoliBox
has exchanged the coefficents of our w state so that the nonzero coefficents are now on the \(000\rangle\) and \(111\rangle\) bitstrings with the coefficent of \(010\rangle\) remaining unchanged.
Importing/Exporting Circuits¶
pytket
Circuit
s can be natively serialized and deserialized from JSONcompatible dictionaries, using the Circuit.to_dict()
and Circuit.from_dict()
methods. This is the method of serialization which supports the largest class of circuits, and provides the highest fidelity.
import tempfile
import json
from pytket import Circuit, OpType
circ = Circuit(2)
circ.Rx(0.1, 0)
circ.CX(0, 1)
circ.YYPhase(0.2, 0, 1)
circ_dict = circ.to_dict()
print(circ_dict)
with tempfile.TemporaryFile('w+') as fp:
json.dump(circ_dict, fp)
fp.seek(0)
new_circ = Circuit.from_dict(json.load(fp))
draw(new_circ)
{'bits': [], 'commands': [{'args': [['q', [0]]], 'op': {'params': ['0.1'], 'type': 'Rx'}}, {'args': [['q', [0]], ['q', [1]]], 'op': {'type': 'CX'}}, {'args': [['q', [0]], ['q', [1]]], 'op': {'params': ['0.2'], 'type': 'YYPhase'}}], 'created_qubits': [], 'discarded_qubits': [], 'implicit_permutation': [[['q', [0]], ['q', [0]]], [['q', [1]], ['q', [1]]]], 'phase': '0.0', 'qubits': [['q', [0]], ['q', [1]]]}
pytket
also supports interoperability with a number of other quantum software frameworks and programming languages for easy conversion of existing code and to provide users the freedom to choose their preferred input system and use available highlevel packages.
OpenQASM is one of the current industry standards for lowlevel circuit description languages, featuring named quantum and classical registers, parameterised subroutines, and a limited form of conditional execution. Having bidirectional conversion support allows this to double up as a method of serializing circuits for later use. Though less expressive than native dictionary serialization, it is widely supported and so serves as a platformindependent method of storing circuits.
from pytket.qasm import circuit_from_qasm, circuit_to_qasm_str
import tempfile, os
fd, path = tempfile.mkstemp(".qasm")
os.write(fd, """OPENQASM 2.0;
include "qelib1.inc";
qreg q[2];
creg c[2];
h q[0];
cx q[0], q[1];
cz q[1], q[0];
measure q > c;
""".encode())
os.close(fd)
circ = circuit_from_qasm(path)
os.remove(path)
print(circuit_to_qasm_str(circ)) # print QASM string
OPENQASM 2.0;
include "qelib1.inc";
qreg q[2];
creg c[2];
h q[0];
cx q[0],q[1];
cz q[1],q[0];
measure q[0] > c[0];
measure q[1] > c[1];
Note
The OpenQASM converters do not support circuits with implicit qubit permutations. This means that if a circuit contains such a permutation it will be ignored when exported to OpenQASM format.
The core pytket
package additionally features a converter from Quipper, another circuit description language.
from pytket.quipper import circuit_from_quipper
import tempfile, os
fd, path = tempfile.mkstemp(".quip")
os.write(fd, """Inputs: 0:Qbit, 1:Qbit, 2:Qbit
QGate["X"](0)
QGate["Y"](1)
QGate["Z"](2)
Outputs: 0:Qbit, 1:Qbit, 2:Qbit
""".encode())
os.close(fd)
circ = circuit_from_quipper(path)
draw(circ)
os.remove(path)
Note
There are a few features of the Quipper language that are not supported by the converter, which are outlined in the pytket.quipper
documentation.
Converters for other quantum software frameworks can optionally be included by installing the corresponding extension module. These are additional PyPI packages with names pytketX
, which extend the pytket
namespace with additional features to interact with other systems, either using them as a frontend for circuit construction and highlevel algorithms or targeting simulators and devices as backends.
For example, installing the pytketqiskit
package will add the tk_to_qiskit()
and qiskit_to_tk()
methods which convert between the Circuit
class from pytket
and QuantumCircuit
.
from qiskit import QuantumCircuit
from math import pi
qc = QuantumCircuit(3)
qc.h(0)
qc.cx(0, 1)
qc.rz(pi/2, 1)
print(qc)
┌───┐
q_0: ┤ H ├──■─────────────
└───┘┌─┴─┐┌─────────┐
q_1: ─────┤ X ├┤ Rz(π/2) ├
└───┘└─────────┘
q_2: ─────────────────────
We can convert this QuantumCircuit
to a pytket Circuit
, append some gates and then convert back.
from pytket.extensions.qiskit import qiskit_to_tk, tk_to_qiskit
circ = qiskit_to_tk(qc)
circ.CX(1, 2)
circ.measure_all()
qc2 = tk_to_qiskit(circ)
print(qc2)
┌───┐ ┌─┐
q_0: ┤ H ├──■──┤M├──────────────────────
└───┘┌─┴─┐└╥┘┌─────────┐ ┌─┐
q_1: ─────┤ X ├─╫─┤ Rz(π/2) ├──■──┤M├───
└───┘ ║ └─────────┘┌─┴─┐└╥┘┌─┐
q_2: ───────────╫────────────┤ X ├─╫─┤M├
║ └───┘ ║ └╥┘
c: 3/═══════════╩══════════════════╩══╩═
0 1 2
Symbolic Circuits¶
In practice, it is very common for an experiment to use many circuits with similar structure but with varying gate parameters. In variational algorithms like VQE and QAOA, we are trying to explore the energy landscape with respect to the circuit parameters, realised as the angles of rotation gates. The only differences between iterations of the optimisation procedure are the specific angles of rotations in the circuits. Because the procedures of generating and compiling the circuits typically won’t care what the exact angles are, we can define the circuits abstractly, treating each parameter as an algebraic symbol. The circuit generation and compilation can then be pulled outside of the optimisation loop, being performed once and for all rather than once for each set of parameter values.
sympy is a widelyused python package for symbolic expressions and algebraic manipulation, defining a sympy Symbol
objects to represent algebraic variables and using them in sympy Expression s to build mathematical statements and arithmetic expressions. Symbolic circuits are managed in pytket
by defining the circuit parameters as sympy.Symbol
s, which can be passed in as arguments to the gates and later substituted for concrete values.
from pytket import Circuit, OpType
from sympy import Symbol
a = Symbol("alpha")
b = Symbol("beta")
circ = Circuit(2)
circ.Rx(a, 0)
circ.Rx(2*a, 1)
circ.CX(0, 1)
circ.YYPhase(b, 0, 1)
draw(circ)
s_map = {a:0.3, b:1.25}
circ.symbol_substitution(s_map)
print(circ.free_symbols())
set()
It is important to note that the units of the parameter values will still be in halfturns, and so may need conversion to/from radians if there is important semantic meaning to the parameter values. This can either be done at the point of interpreting the values, or by embedding the conversion into the Circuit
.
from pytket import Circuit
from sympy import Symbol, pi
a = Symbol("alpha") # suppose that alpha is given in radians
circ = Circuit(2) # convert alpha to halfturns when adding gates
circ.Rx(a/pi, 0).CX(0, 1).Ry(a/pi, 0)
s_map = {a: pi/4}
circ.symbol_substitution(s_map)
draw(circ)
Substitution need not be for concrete values, but is defined more generally to allow symbols to be replaced by arbitrary expressions, including other symbols. This allows for alphaconversion or to look at special cases with redundant parameters.
from pytket import Circuit
from sympy import symbols
a, b, c = symbols("a b c")
circ = Circuit(2)
circ.Rx(a, 0).Rx(b, 1).CX(0, 1).Ry(c, 0).Ry(c, 1)
s_map = {a: 2*a, c: a} # replacement happens simultaneously, and not recursively
circ.symbol_substitution(s_map)
draw(circ)
There are currently no simulators or devices that can run symbolic circuits algebraically, so every symbol must be instantiated before running. At any time, you can query the Circuit
object for the set of free symbols it contains to check what would need to be instantiated before it can be run.
from pytket import Circuit
from sympy import symbols
a, b = symbols("a, b")
circ = Circuit(2)
circ.Rx(a, 0).Rx(b, 1).CZ(0, 1)
circ.symbol_substitution({a:0.2})
print(circ.free_symbols())
print(circ.is_symbolic()) # returns True when free_symbols() is nonempty
{b}
True
Note
There are some minor drawbacks associated with symbolic compilation. When using Eulerangle equations or quaternions for merging adjacent rotation gates, the resulting angles are given by some lengthy trigonometric expressions which cannot be evaluated down to just a number when one of the original angles was parameterised; this can lead to unhelpfully long expressions for the angles of some gates in the compiled circuit. It is also not possible to apply the pytket.passes.KAKDecomposition
pass to simplify a parameterised circuit, so that pass will only apply to nonparameterised subcircuits, potentially missing some valid opportunities for optimisation.
See also
To see how to use symbolic compilation in a variational experiment, have a look at our VQE (UCCSD) example.
Symbolic unitaries and states¶
In pytket.utils.symbolic
we provide functions circuit_to_symbolic_unitary()
, which can calculate the unitary representation of a possibly symbolic circuit, and circuit_apply_symbolic_statevector()
, which can apply a symbolic circuit to an input statevector and return the output state (effectively simulating it).
from pytket.utils.symbolic import circuit_apply_symbolic_statevector, circuit_to_symbolic_unitary
from sympy import Symbol, pi
a = Symbol("alpha")
circ = Circuit(2)
circ.Rx(a/pi, 0).CX(0, 1)
# All zero input state is assumed if no initial state is provided
display(circuit_apply_symbolic_statevector(circ))
circuit_to_symbolic_unitary(circ)
The unitaries are calculated using the unitary representation of each OpType , and according to the default ILO BasisOrder convention used in backends ILO BasisOrder convention used in backends.
The outputs are sympy ImmutableMatrix objects, and use the same symbols as in the circuit, so can be further substituted and manipulated.
The conversion functions use the sympy Quantum Mechanics module, see also the circuit_to_symbolic_gates()
and circuit_apply_symbolic_qubit()
functions to see how to work with those objects directly.
Warning
Unitaries corresponding to circuits with \(n\) qubits have dimensions \(2^n \times 2^n\), so are computationally very expensive to calculate. Symbolic calculation is also computationally costly, meaning calculation of symbolic unitaries is only really feasible for very small circuits (of up to a few qubits in size). These utilities are provided as way to test the design of small subcircuits to check they are performing the intended unitary. Note also that as mentioned above, compilation of a symbolic circuit can generate long symbolic expressions; converting these circuits to a symbolic unitary could then result in a matrix object that is very hard to work with or interpret.
Advanced Circuit Construction Topics¶
Custom parameterised Gates¶
The CircBox
construction is good for subroutines where the instruction sequence is fixed. The CustomGateDef
construction generalises this to construct parameterised subroutines by binding symbols in the definition circuit and instantiating them at each instance. Any symbolic Circuit
can be provided as the subroutine definition. Remaining symbols that are not bound are treated as free symbols in the global scope.
from pytket.circuit import Circuit, CustomGateDef
from sympy import symbols
a, b = symbols("a b")
def_circ = Circuit(2)
def_circ.CZ(0, 1)
def_circ.Rx(a, 1)
def_circ.CZ(0, 1)
def_circ.Rx(a, 1)
def_circ.Rz(b, 0)
gate_def = CustomGateDef.define("MyCRx", def_circ, [a])
circ = Circuit(3)
circ.add_custom_gate(gate_def, [0.2], [0, 1])
circ.add_custom_gate(gate_def, [0.3], [0, 2])
draw(circ)
print(circ.free_symbols()) # Print remaining free symbols
{b}
Clifford Tableaux¶
The Clifford (a.k.a. stabilizer) fragment of quantum theory is known to exhibit efficient classical representations of states and unitaries. This allows for short descriptions that can fully characterise a state/unitary and efficient circuit simulation. Whilst the Clifford group can be characterised as the operations generated by CX
, H
, and S
gates with qubit initialisation in the \(0\rangle\) state, it is also the largest group of operations under which the Pauli group is closed, i.e. for any tensor of Paulis \(P\) and Clifford operation \(C\), \(CPC^\dagger\) is also a tensor of Paulis.
Any state \(\psi\rangle\) in the Clifford fragment is uniquely identified by those tensors of Pauli operators that stabilize it (those \(P\) such that \(P\psi\rangle = \psi\rangle\)). These stabilizers form a group of size \(2^n\) for an \(n\) qubit state, but it is sufficient to identify \(n\) independent generators to specify the group. If a Clifford gate \(C\) is applied to the state, each generator \(P\) can be updated to \(P' = CPC^\dagger\) since \(C\psi\rangle = CP\psi\rangle = (CPC^\dagger)C\psi\rangle\). We can therefore characterise each Clifford operation by its actions on generators of the Pauli group, giving us the Clifford tableau form. In pytket
, the UnitaryTableau
class uses the binary symplectic representation from Aaronson and Gottesman [7].
from pytket.circuit import OpType, Qubit
from pytket.tableau import UnitaryTableau
tab = UnitaryTableau(3)
tab.apply_gate_at_end(OpType.S, [Qubit(0)])
tab.apply_gate_at_end(OpType.CX, [Qubit(1), Qubit(2)])
print(tab)
X@q[0] > 1 0 0 1 0 0 0
X@q[1] > 0 1 1 0 0 0 0
X@q[2] > 0 0 1 0 0 0 0

Z@q[0] > 0 0 0 1 0 0 0
Z@q[1] > 0 0 0 0 1 0 0
Z@q[2] > 0 0 0 0 1 1 0
The way to interpret this format is that, for example, the top rows state that the unitary transforms \(X_0 I_1 I_2\) at its input to \(Y_0 I_1 I_2\) at its output, and it transforms \(I_0 X_1 I_2\) to \(I_0 X_1 X_2\).
The primary use for tableaux in pytket
is as a scalable means of specifying a Clifford unitary for insertion into a circuit as a Box. This can then be decomposed into basic gates during compilation.
from pytket.circuit import Circuit
from pytket.tableau import UnitaryTableauBox
box = UnitaryTableauBox(
np.asarray([[1, 1, 0], [0, 1, 0], [0, 0, 1]], dtype=bool),
np.asarray([[0, 0, 0], [0, 0, 0], [0, 0, 1]], dtype=bool),
np.asarray([0, 0, 1], dtype=bool),
np.asarray([[0, 0, 0], [0, 1, 0], [0, 0, 0]], dtype=bool),
np.asarray([[1, 0, 0], [1, 1, 0], [0, 0, 1]], dtype=bool),
np.asarray([1, 0, 1], dtype=bool)
)
circ = Circuit(3)
circ.add_gate(box, [0, 1, 2])
draw(circ)
After the tableau is added to a circuit, it can be readily decomposed to Clifford gates.
from pytket.passes import DecomposeBoxes, RemoveRedundancies
DecomposeBoxes().apply(circ)
RemoveRedundancies().apply(circ) # Eliminate some redundant gates
draw(circ)
Note
The current decomposition method for tableaux is not particularly efficient in terms of gate count, so consider using higher optimisation levels when compiling to help reduce the gate cost.
The data structure used here for tableaux is intended for compilation use. For fast simulation of Clifford circuits, we recommend using the StimBackend
from pytketstim
, the SimplexBackend
from pytketpysimplex
(optimized for large sparse circuits), or the AerBackend
from pytketqiskit
. Future versions of pytket
may include improved decompositions from tableaux, as well as more flexible tableaux to represent stabilizer states, isometries, and diagonalisation circuits.
Classical and conditional operations¶
Moving beyond toy circuit examples, many applications of quantum computing require looking at circuits as POVMs for extra expressivity, or introducing errorcorrecting schemes to reduce the effective noise. Each of these requires performing measurements midcircuit and then performing subsequent gates conditional on the classical value of the measurement result, or on the results of calculations on the results.
Any pytket
operation can be made conditional at the point of adding it to
the Circuit
by providing the condition
kwarg. The interpretation
of circ.G(q, condition=reg[0])
is: “if the bit reg[0]
is set to 1, then
perform G(q)
”.
Conditions on more complicated expressions over the values of Bit
and BitRegister
are also
possible, expressed as conditions on the results of expressions involving
bitwise AND (&), OR () and XOR (^) operations. In the case of registers, you
can also express arithmetic operations: add (+), subtract (), multiply (*),
floor/integer division (//), left shift (<<) and right shift (>>).
For example a gate can be made conditional on the result of a bitwise XOR of
registers a
, b
, and c
being larger than 4 by writing circ.G(q, condition=reg_gt(a ^ b ^ c, 4))
.
When such a condition is added, the result of the expression is written to a
scratch bit or register, and the gate is made conditional on the value of the
scratch variable.
For comparison of registers, a special RangePredicate
type is used to encode
the result of the comparison onto a scratch bit.
See the pytket.circuit.logic_exp
documentation for more on the
possible expressions and predicates.
from pytket.circuit import (
Circuit,
BitRegister,
if_bit,
if_not_bit,
reg_eq,
reg_geq,
reg_gt,
reg_leq,
reg_lt,
reg_neq,
)
# create a circuit and add quantum and classical registers
circ = Circuit()
qreg = circ.add_q_register("q", 10)
reg_a = circ.add_c_register("a", 4)
# another way of adding a register to the Circuit
reg_b = BitRegister("b", 3)
circ.add_c_register(reg_b)
reg_c = circ.add_c_register("c", 3)
# if (reg_a[0] == 1)
circ.H(qreg[0], condition=reg_a[0])
circ.X(qreg[0], condition=if_bit(reg_a[0]))
# if (reg_a[2] == 0)
circ.T(qreg[1], condition=if_not_bit(reg_a[2]))
# compound logical expressions
circ.Z(qreg[0], condition=(reg_a[2] & reg_a[3]))
circ.Z(qreg[1], condition=if_not_bit(reg_a[2] & reg_a[3]))
big_exp = reg_a[0]  reg_a[1] ^ reg_a[2] & reg_a[3]
# syntactic sugar for big_exp = BitOr(reg_a[0], BitXor(reg_a[1], BitAnd(reg_a[2], reg_a[3])))
circ.CX(qreg[1], qreg[2], condition=big_exp)
# Register comparisons
# if (reg_a == 3)
circ.H(qreg[2], condition=reg_eq(reg_a, 3))
# if (reg_c != 6)
circ.Y(qreg[4], condition=reg_neq(reg_c, 5))
# if (reg_b < 6)
circ.X(qreg[3], condition=reg_lt(reg_b, 6))
# if (reg_b > 3)
circ.Z(qreg[5], condition=reg_gt(reg_b, 3))
# if (reg_c <= 6)
circ.S(qreg[6], condition=reg_leq(reg_c, 6))
# if (reg_a >= 3)
circ.T(qreg[7], condition=reg_geq(reg_a, 3))
# compound register expressions
big_reg_exp = (reg_a & reg_b)  reg_c
circ.CX(qreg[3], qreg[4], condition=reg_eq(big_reg_exp, 3))
[IF ([a[0]] == 1) THEN H q[0]; IF ([a[2]] == 0) THEN T q[1]; ClassicalExpBox a[2], a[3], tk_SCRATCH_BIT[0]; ClassicalExpBox a[2], a[3], tk_SCRATCH_BIT[1]; ClassicalExpBox a[0], a[1], a[2], a[3], tk_SCRATCH_BIT[2]; RangePredicate([3,3]) a[0], a[1], a[2], a[3], tk_SCRATCH_BIT[3]; RangePredicate([5,5]) c[0], c[1], c[2], tk_SCRATCH_BIT[4]; RangePredicate([0,5]) b[0], b[1], b[2], tk_SCRATCH_BIT[5]; RangePredicate([4,18446744073709551615]) b[0], b[1], b[2], tk_SCRATCH_BIT[6]; RangePredicate([0,6]) c[0], c[1], c[2], tk_SCRATCH_BIT[7]; RangePredicate([3,18446744073709551615]) a[0], a[1], a[2], a[3], tk_SCRATCH_BIT[8]; ClassicalExpBox a[0], a[1], a[2], a[3], b[0], b[1], b[2], c[0], c[1], c[2], tk_SCRATCH_BITREG_0[0], tk_SCRATCH_BITREG_0[1], tk_SCRATCH_BITREG_0[2]; IF ([a[0]] == 1) THEN X q[0]; IF ([tk_SCRATCH_BIT[1]] == 0) THEN Z q[1]; IF ([tk_SCRATCH_BIT[5]] == 1) THEN X q[3]; IF ([tk_SCRATCH_BIT[4]] == 0) THEN Y q[4]; IF ([tk_SCRATCH_BIT[6]] == 1) THEN Z q[5]; IF ([tk_SCRATCH_BIT[7]] == 1) THEN S q[6]; IF ([tk_SCRATCH_BIT[8]] == 1) THEN T q[7]; RangePredicate([3,3]) tk_SCRATCH_BITREG_0[0], tk_SCRATCH_BITREG_0[1], tk_SCRATCH_BITREG_0[2], tk_SCRATCH_BIT[9]; IF ([tk_SCRATCH_BIT[0]] == 1) THEN Z q[0]; IF ([tk_SCRATCH_BIT[2]] == 1) THEN CX q[1], q[2]; IF ([tk_SCRATCH_BIT[9]] == 1) THEN CX q[3], q[4]; IF ([tk_SCRATCH_BIT[3]] == 1) THEN H q[2]; ]
So far we’ve looked at conditioning the application of a gate on bits,
registers, or expressions over those. We can also write some more standard
classical computations by assigning the result of some computation to output
bits or registers. We can also set the value or copy the contents of one resource
in to another. Note in the examples below to express something like <var> = <exp>
we use circuit methods (like add_c_setreg()
, or
add_classicalexpbox_register()
) that take <exp>
as the first input and <var>
as the second. Note that these classical operations can be conditional on other
classical operations, just like quantum operations.
from pytket.circuit import Circuit, reg_gt
# create a circuit and add some classical registers
circ = Circuit()
reg_a = circ.add_c_register("a", 4)
reg_b = circ.add_c_register("b", 3)
reg_c = circ.add_c_register("c", 3)
# Write to classical registers
# a = 3
circ.add_c_setreg(3, reg_a)
# a[0] = 1
circ.add_c_setbits([1], [reg_a[0]])
# Copy: b = a
# b is smaller than a so the first 3 bits of a will be copied
circ.add_c_copyreg(reg_a, reg_b)
# b[1] = a[2]
circ.add_c_copybits([reg_a[2]], [reg_b[1]])
# Conditional classical operation
# if (a > 1) b = 3
circ.add_c_setreg(3, reg_b, condition=reg_gt(reg_a, 1))
# Write out the results of logical expressions
# c = a ^ b
circ.add_classicalexpbox_register(reg_a ^ reg_b, reg_c)
# c[0] = a[1] & b[2]
circ.add_classicalexpbox_bit(reg_a[1] & reg_b[2], [reg_c[0]])
# Register arithmetic
# c = a + b // c (note the use of the floor divide symbol)
circ.add_classicalexpbox_register(reg_a + reg_b // reg_c, reg_c)
# a = a  b * c
circ.add_classicalexpbox_register(reg_a  reg_b * reg_c, reg_a)
# a = a << 2
circ.add_classicalexpbox_register(reg_a << 2, reg_a)
# c = b >> 1
circ.add_classicalexpbox_register(reg_b >> 1, reg_c)
[SetBits(1100) a[0], a[1], a[2], a[3]; SetBits(1) a[0]; CopyBits a[0], a[1], a[2], b[0], b[1], b[2]; RangePredicate([2,18446744073709551615]) a[0], a[1], a[2], a[3], tk_SCRATCH_BIT[0]; CopyBits a[2], b[1]; IF ([tk_SCRATCH_BIT[0]] == 1) THEN SetBits(110) b[0], b[1], b[2]; ClassicalExpBox a[0], a[1], a[2], a[3], b[0], b[1], b[2], c[0], c[1], c[2]; ClassicalExpBox a[1], b[2], c[0]; ClassicalExpBox a[0], a[1], a[2], a[3], b[0], b[1], b[2], c[0], c[1], c[2]; ClassicalExpBox b[0], b[1], b[2], c[0], c[1], c[2], a[0], a[1], a[2], a[3]; ClassicalExpBox a[0], a[1], a[2], a[3]; ClassicalExpBox b[0], b[1], b[2], c[0], c[1], c[2]; ]
Warning
Unlike most uses of readouts in pytket
, register comparisons expect a littleendian value, e.g. in the above example condition=reg_eq(reg_a, 3)
(representing the littleendian binary string 110000...
) is triggered when reg_a[0]
and reg_a[1]
are in state 1
and the remainder of the register is in state 0
.
Note
This feature is only usable on a limited selection of devices and simulators which support conditional gates or classical operations.
The AerBackend
(from pytketqiskit) can support the OpenQasm model,
where gates can only be conditional on an entire classical register being an
exact integer value. Bitwise logical operations and register arithmetic are not supported.
Therefore only conditions of the form
condition=reg_eq(reg, val)
are valid.
The QuantinuumBackend
(from pytketquantinuum)
can support the full range of expressions and comparisons shown above.
CircuitLevel Operations¶
Systematic modifications to a Circuit
object can go beyond simply adding gates one at a time. For example, given a unitary Circuit
, we may wish to generate its inverse for the purposes of uncomputation of ancillae or creating conjugation circuits to diagonalise an operator as in the sample below.
from pytket import Circuit
# we want a circuit for E = exp(i pi (0.3 XX + 0.1 YY))
circ = Circuit(2)
# find C such that C; Rx(a, 0); C^dagger performs exp(i a pi XX/2)
# and C; Rz(b, 1); C^dagger performs exp(i b pi YY/2)
conj = Circuit(2)
conj.V(0).V(1).CX(0, 1)
conj_dag = conj.dagger()
circ.append(conj)
circ.Rx(0.6, 0).Rz(0.2, 1)
circ.append(conj_dag)
Generating the transpose of a unitary works similarly using transpose()
.
Note
Since it is not possible to construct the inverse of an arbitrary POVM, the dagger()
and transpose()
methods will fail if there are any measurements, resets, or other operations that they cannot directly invert.
Implicit Qubit Permutations¶
The Circuit
class is built as a DAG to help follow the paths of resources and represent the circuit canonically up to trivial commutations. Each of the edges represents a resource passing from one instruction to the next, so we could represent SWAPs (and general permutations) by connecting the predecessors of the SWAP instruction to the opposite successors. This eliminates the SWAP instruction from the graph (meaning we would no longer perform the operation at runtime) and could enable the compiler to spot additional opportunities for simplification. One example of this in practice is the ability to convert a pair of CXs in opposite directions to just a single CX (along with an implicit SWAP that isn’t actually performed).
from pytket import Circuit
from pytket.utils import Graph
circ = Circuit(4)
circ.CX(0, 1)
circ.CX(1, 0)
circ.Rx(0.2, 1)
circ.CZ(0, 1)
print(circ.get_commands())
Graph(circ).get_DAG()
[CX q[0], q[1];, CX q[1], q[0];, Rx(0.2) q[1];, CZ q[0], q[1];]
from pytket.passes import CliffordSimp
CliffordSimp().apply(circ)
print(circ.get_commands())
print(circ.implicit_qubit_permutation())
Graph(circ).get_DAG()
[TK1(1.5, 2.2, 0) q[0];, TK1(0, 0, 0.5) q[1];, CX q[1], q[0];, TK1(0, 0, 0.5) q[0];]
{q[0]: q[1], q[1]: q[0], q[2]: q[2], q[3]: q[3]}
This procedure essentially exploits the naturality of the symmetry operator in the resource theory to push it to the end of the circuit: the Rx
gate has moved from qubit q[1]
to q[0]
and can be commuted through to the start. This is automatically considered when composing two Circuit
s together.
The permutation has been reduced to something implicit in the graph, and we now find that tracing a path from an input can reach an output with a different UnitID
. Since this permutation is missing in the command sequence, simulating the circuit would only give the correct state up to a permutation of the qubits. This does not matter when running on real devices where the final quantum system is discarded after use, but is detectable when using a statevector simulator. This is handled automatically by pytket
backends, but care should be taken when reading from the Circuit
directly  two quantum Circuit
s can have the same sequence of instructions but different unitaries because of implicit permutations. This permutation information is typically dropped when exporting to another software framework. The implicit_qubit_permutation()
method can be used to inspect such a permutation.
Modifying Operations Within Circuits¶
Symbolic parameters allow one to construct a circuit with some notyetassigned parameters, and later (perhaps after some optimization), to instantiate them with different values. Occasionally, however, one may desire more flexibility in substituting operations within a circuit. For example, one may wish to apply controls from a certain qubit to certain operations, or to insert or remove certain operations.
This can be achieved with pytket
, provided the mutable operations are tagged during circuit construction with identifying names (which can be arbitrary strings). If two operations are given the same name then they belong to the same “operation group”; they can (and must) then be substituted simultaneously.
Both primitive gates and boxes can be tagged and substituted in this way. The only constraint is that the signature (number and order of quantum and classical wires) of the substituted operation must match that of the original operation in the circuit. (It follows that all operations in the same group must have the same signature. An attempt to add an operation with an existing name with a mismatching signature will fail.)
To add gates or boxes to a circuit with specified op group names, simply pass the name as a keyword argument opgroup
to the method that adds the gate or box. To substitute all operations in a group, use the substitute_named()
method. This can be used to substitute a circuit, an operation or a box into the existing circuit.
from pytket.circuit import Circuit, CircBox
circ = Circuit(3)
circ.Rz(0.25, 0, opgroup="rotations")
circ.CX(0, 1)
circ.Ry(0.75, 1, opgroup="rotations")
circ.H(2, opgroup="special one")
circ.CX(2, 1)
cbox = CircBox(Circuit(2, name="P").S(0).CY(0, 1))
circ.add_gate(cbox, [0, 1], opgroup="Fred")
circ.CX(1, 2, opgroup="Fred")
draw(circ)
from pytket.circuit import Op
# Substitute a new 1qubit circuit for all ops in the "rotations" group:
newcirc = Circuit(1).Rx(0.125, 0).Ry(0.875, 0)
circ.substitute_named(newcirc, "rotations")
# Replace the "special one" with a different op:
newop = Op.create(OpType.T)
circ.substitute_named(newop, "special one")
# Substitute a box for the "Fred" group:
newcbox = CircBox(Circuit(2, name="Q").H(1).CX(1, 0))
circ.substitute_named(newcbox, "Fred")
draw(circ)
Note that when an operation or box is substituted in, the op group name is retained (and further substitutions can be made). When a circuit is substituted in, the op group name disappears.
To remove an operation, one can replace it with an empty circuit.
To add a control to an operation, one can add the original operation as a CircBox
with one unused qubit, and subtitute it with a QControlBox
.
from pytket.circuit import QControlBox
def with_empty_qubit(op: Op) > CircBox:
n_qb = op.n_qubits
return CircBox(Circuit(n_qb + 1).add_gate(op, list(range(1, n_qb + 1))))
def with_control_qubit(op: Op) > QControlBox:
return QControlBox(op, 1)
c = Circuit(3)
h_op = Op.create(OpType.H)
cx_op = Op.create(OpType.CX)
h_0_cbox = with_empty_qubit(h_op)
h_q_qbox = with_control_qubit(h_op)
cx_0_cbox = with_empty_qubit(cx_op)
cx_q_qbox = with_control_qubit(cx_op)
c.X(0).Y(1)
c.add_gate(h_0_cbox, [2, 0], opgroup="hgroup")
c.add_gate(cx_0_cbox, [2, 0, 1], opgroup="cxgroup")
c.Y(0).X(1)
c.add_gate(h_0_cbox, [2, 1], opgroup="hgroup")
c.add_gate(cx_0_cbox, [2, 1, 0], opgroup="cxgroup")
c.X(0).Y(1)
c.substitute_named(h_q_qbox, "hgroup")
c.substitute_named(cx_q_qbox, "cxgroup")
draw(c)