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Example: Gate synthesis

This example demonstrates how the built-in compiler can be used to perform gate synthesis.

Synthesizing Single-Qubit Gates

We begin by generating a random gate in \(SU(2)\); we will synthesize this gate later.

import trueq as tq

# create a Haar random SU(2) gate and print its matrix representation
U = tq.Gate.random(2)
U

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Name:

• Gate(Y, X, ...)

Generators:

• 'Y': 100.341
• 'X': -129.067
• 'Z': -24.005

Matrix:

We can perform single qubit gate decomposition into a number of possible “modes”. Here we decompose the gate U into a ZXZXZ decomposition, which is short hand for \(Z(\theta)X(90)Z(\phi)X(90)Z(\gamma)\). See QubitMode for a complete list of all available single qubit decompositions.

synthesized_gate = tq.math.QubitMode.ZXZXZ.decompose(U)

# print the synthesized gate as a list of single-qubit rotations about Z and X
synthesized_gate

[('Z', 79.58855658027758),
 ('X', 90),
 ('Z', 22.26079950862166),
 ('X', 90),
 ('Z', 3.863045721614313)]

Synthesizing Two-Qubit Gates

In the event that we want to express a two-qubit gate in terms of a different two-qubit gate, we can use the compiler to synthesize the desired gate. Here we decompose a random \(SU(4)\) operation so that it can be implemented using iSWAP gates.

# define the gate to be synthesized
gate_to_be_synthesized = tq.Gate.random(4)

# re-express the gate using an iswap gate as the two-qubit gate
two_qubit_synthesized_gate = tq.math.decompose_unitary(
    target_gate=gate_to_be_synthesized, given_gate=tq.Gate.iswap
)

# print the synthesized gate
two_qubit_synthesized_gate

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Circuit Key: No key present in circuit.
Marker 0 Compilation tools may only recompile cycles with equal markers.	(0): Gate(Y, X, ...) Name: Gate(Y, X, ...) Generators: 'Y': -121.294 'X': -116.259 'Z': 51.663 Matrix:	(1): Gate(Y, X, ...) Name: Gate(Y, X, ...) Generators: 'Y': -160.54 'X': 37.359 'Z': -104.788 Matrix:
Marker 0 Compilation tools may only recompile cycles with equal markers.	(0, 1): Gate.iswap Name: Gate.iswap Aliases: Gate.iswap Likeness: iSWAP Generators: 'YY': -90.0 'XX': -90.0 Matrix:
Marker 0 Compilation tools may only recompile cycles with equal markers.	(0): Gate(Y, X, ...) Name: Gate(Y, X, ...) Generators: 'Y': 37.385 'X': 74.107 'Z': 92.372 Matrix:	(1): Gate(Y, X, ...) Name: Gate(Y, X, ...) Generators: 'Y': 27.658 'X': -22.209 'Z': -44.033 Matrix:
Marker 0 Compilation tools may only recompile cycles with equal markers.	(0, 1): Gate.iswap Name: Gate.iswap Aliases: Gate.iswap Likeness: iSWAP Generators: 'YY': -90.0 'XX': -90.0 Matrix:
Marker 0 Compilation tools may only recompile cycles with equal markers.	(0): Gate(Y, X, ...) Name: Gate(Y, X, ...) Generators: 'Y': 6.604 'X': 28.894 'Z': -134.88 Matrix:	(1): Gate(Y, X, ...) Name: Gate(Y, X, ...) Generators: 'Y': 127.755 'X': -2.436 'Z': 123.108 Matrix:
Marker 0 Compilation tools may only recompile cycles with equal markers.	(0, 1): Gate.iswap Name: Gate.iswap Aliases: Gate.iswap Likeness: iSWAP Generators: 'YY': -90.0 'XX': -90.0 Matrix:
Marker 0 Compilation tools may only recompile cycles with equal markers.	(0): Gate(Y, X, ...) Name: Gate(Y, X, ...) Generators: 'Y': -61.493 'X': -18.753 'Z': 210.802 Matrix:	(1): Gate(Y, X, ...) Name: Gate(Y, X, ...) Generators: 'Y': 27.753 'X': 73.269 'Z': 177.858 Matrix:

This circuit can be verified to reproduce the original random unitary using an ideal simulator:

matrix = tq.Simulator().operator(two_qubit_synthesized_gate).mat()

# This will result in an identity gate up to a global complex phase.
tq.plot_mat(matrix @ gate_to_be_synthesized.adj.mat)

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