# Symbolic compilation¶

Download this notebook - symbolics_example.ipynb

Motivation: in compilation, particularly of hybrid classical-quantum variational algorithms in which the structure of a circuit remains constant but the parameters of some gates change, it can be useful to compile using symbolic parameters and optimise the circuit without knowledge of what these parameters will be instantiated to afterwards.

In this tutorial, we will show how to compile a circuit containing mathematical symbols, and then instantiate the symbols afterwards. To do this, you need to have pytket installed. Run:

pip install pytket

To begin, we will import the Circuit and Transform classes from pytket, and the fresh_symbol method from pytket.circuit.

from pytket.circuit import Circuit, fresh_symbol
from pytket.transform import Transform


Now, we can construct a circuit containing symbols. You can ask for symbols by calling the fresh_symbol method with a string as an argument. This string represents the preferred symbol name; if this has already been used elsewhere, an appropriate suffix of the form _x, with x a natural number, will be added to generate a new symbol, as shown below:

a = fresh_symbol("a")
a1 = fresh_symbol("a")
print(a)
print(a1)

a
a_1


We are going to make a circuit using just three ‘phase gadgets’: Rz gates surrounded by ladders of CX gates.

b = fresh_symbol("b")
circ = Circuit(4)
circ.CX(0, 1)
circ.CX(1, 2)
circ.CX(2, 3)
circ.Rz(a, 3)
circ.CX(2, 3)
circ.CX(1, 2)
circ.CX(0, 1)
circ.CX(3, 2)
circ.CX(2, 1)
circ.CX(1, 0)
circ.Rz(b, 0)
circ.CX(1, 0)
circ.CX(2, 1)
circ.CX(3, 2)
circ.CX(0, 1)
circ.CX(1, 2)
circ.CX(2, 3)
circ.Rz(0.5, 3)
circ.CX(2, 3)
circ.CX(1, 2)
circ.CX(0, 1)

[CX q[0], q[1]; CX q[1], q[2]; CX q[2], q[3]; Rz(a) q[3]; CX q[2], q[3]; CX q[1], q[2]; CX q[0], q[1]; CX q[3], q[2]; CX q[2], q[1]; CX q[1], q[0]; Rz(b) q[0]; CX q[1], q[0]; CX q[2], q[1]; CX q[0], q[1]; CX q[3], q[2]; CX q[1], q[2]; CX q[2], q[3]; Rz(0.5) q[3]; CX q[2], q[3]; CX q[1], q[2]; CX q[0], q[1]; ]


Now we can use the render_circuit_jupyter method to display the circuit.

from pytket.circuit.display import render_circuit_jupyter

render_circuit_jupyter(circ)


Now let’s use a transform to shrink the circuit. For more detail on transforms, see the transform_example notebook.

Transform.OptimisePhaseGadgets().apply(circ)
render_circuit_jupyter(circ)


Note that the type of gate has changed to U1, but the phase gadgets have been successfully combined. The U1 gate is an IBM-specific gate that is equivalent to an Rz.

We can now instantiate the symbols with some desired values. We make a dictionary, with each key a symbol name, and each value a double. Note that this value is in units of ‘half-turns’, in which a value of $$1$$ corresponds to a rotation of $$\pi$$.

Before instantiating our parameters we make a copy of the circuit, so that we can repeat the exercise without the need for recompilation.

symbol_circ = circ.copy()

symbol_dict = {a: 0.5, b: 0.75}
circ.symbol_substitution(symbol_dict)

render_circuit_jupyter(circ)


Because this symbol substitution was called on the copy, we still have our original symbolic circuit.

render_circuit_jupyter(symbol_circ)


Note: the expression tree for this symbolic expression is very small, consisting of only a couple of different operations, but tket is capable of handling large and complex expressions containing many different types of operation, such as trigonometric functions.

It is usually possible to instantiate a symbolic circuit with specific values that allow further optimisation: for example, if we had chosen $$a=1.5$$ and $$b=0$$, this circuit would be reduce to the identity. If there are likely to be many parameters set to trivial values (such as $$0$$ or $$1$$), it can be beneficial to perform further optimisation after instantiation.