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(X, Y, ...)
Generators:
  • 'X': 105.087
  • 'Y': 81.282
  • 'Z': 17.215
Matrix:
  • 0.40 0.06j -0.19 -0.89j 0.82 -0.41j 0.30 0.28j


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

Out:

[('Z', -110.89189624501294), ('X', 90), ('Z', 48.223321798518754), ('X', 90), ('Z', -35.44956525321924)]

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(X, Y, ...)
Name:
  • Gate(X, Y, ...)
Generators:
  • 'X': -148.468
  • 'Y': 106.714
  • 'Z': 51.023
Matrix:
  • -0.27 -0.07j 0.35 0.89j 0.96 -0.05j 0.18 0.22j
(1): Gate(X, Y, ...)
Name:
  • Gate(X, Y, ...)
Generators:
  • 'X': 2.421
  • 'Y': 19.819
  • 'Z': -69.748
Matrix:
  • -0.13 0.98j -0.06 -0.15j 0.09 0.13j 0.88 0.45j
 
Marker 0
Compilation tools may only recompile cycles with equal markers.
(0, 1): Gate.iswap
Name:
  • Gate.iswap
Aliases:
  • Gate.iswap
Likeness:
  • iSWAP
Generators:
  • 'XX': -90.0
  • 'YY': -90.0
Matrix:
  • 1.00 1.00j 1.00j 1.00
 
 
Marker 0
Compilation tools may only recompile cycles with equal markers.
(0): Gate(X, Y, ...)
Name:
  • Gate(X, Y, ...)
Generators:
  • 'X': -155.59
  • 'Y': -0.706
  • 'Z': 55.909
Matrix:
  • -0.29 -0.21j 0.90 0.25j 0.90 0.25j 0.36 -0.03j
(1): Gate(X, Y, ...)
Name:
  • Gate(X, Y, ...)
Generators:
  • 'X': -73.345
  • 'Y': -58.782
  • 'Z': -156.683
Matrix:
  • 0.28 0.81j 0.44 0.26j -0.16 0.49j -0.32 -0.79j
 
Marker 0
Compilation tools may only recompile cycles with equal markers.
(0, 1): Gate.iswap
Name:
  • Gate.iswap
Aliases:
  • Gate.iswap
Likeness:
  • iSWAP
Generators:
  • 'XX': -90.0
  • 'YY': -90.0
Matrix:
  • 1.00 1.00j 1.00j 1.00
 
 
Marker 0
Compilation tools may only recompile cycles with equal markers.
(0): Gate(X, Y, ...)
Name:
  • Gate(X, Y, ...)
Generators:
  • 'X': -2.787
  • 'Y': -103.925
  • 'Z': 143.133
Matrix:
  • 0.43 -0.68j 0.50 0.31j -0.51 -0.28j -0.39 0.71j
(1): Gate(X, Y, ...)
Name:
  • Gate(X, Y, ...)
Generators:
  • 'X': 43.41
  • 'Y': 130.881
  • 'Z': -32.311
Matrix:
  • 0.15 0.36j -0.56 -0.73j 0.88 0.25j 0.39 0.01j
 
Marker 0
Compilation tools may only recompile cycles with equal markers.
(0, 1): Gate.iswap
Name:
  • Gate.iswap
Aliases:
  • Gate.iswap
Likeness:
  • iSWAP
Generators:
  • 'XX': -90.0
  • 'YY': -90.0
Matrix:
  • 1.00 1.00j 1.00j 1.00
 
 
Marker 0
Compilation tools may only recompile cycles with equal markers.
(0): Gate(X, Y, ...)
Name:
  • Gate(X, Y, ...)
Generators:
  • 'X': 6.295
  • 'Y': 152.905
  • 'Z': -34.54
Matrix:
  • 0.21 -0.21j 0.01 0.96j -0.09 -0.95j -0.23 -0.19j
(1): Gate(X, Y, ...)
Name:
  • Gate(X, Y, ...)
Generators:
  • 'X': 55.775
  • 'Y': -120.955
  • 'Z': 182.215
Matrix:
  • -0.72 -0.44j 0.31 -0.45j -0.54 0.06j 0.05 0.84j


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.visualization.plot_mat(matrix @ gate_to_be_synthesized.adj.mat)
synthesis

Total running time of the script: ( 0 minutes 0.090 seconds)

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