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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.

[2]:
import trueq as tq

# create a Haar random SU(2) gate and print its matrix representation
U = tq.Gate.random(2)
U
[2]:
True-Q formatting will not be loaded without trusting this notebook or rerunning the affected cells. Notebooks can be marked as trusted by clicking "File -> Trust Notebook".
Name:
  • Gate(Y, X, ...)
Generators:
  • 'Y': 121.927
  • 'X': 84.562
  • 'Z': -111.051
Matrix:
  • 0.09 0.59j -0.75 -0.29j 0.53 -0.60j -0.18 -0.57j

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.

[3]:
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
[3]:
[('Z', -239.70358102766073),
 ('X', 90),
 ('Z', 73.78934886465902),
 ('X', 90),
 ('Z', -129.18957320035088)]

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.

[4]:
# 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
[4]:
True-Q formatting will not be loaded without trusting this notebook or rerunning the affected cells. Notebooks can be marked as trusted by clicking "File -> Trust Notebook".
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': -39.915
  • 'X': -107.191
  • 'Z': -42.005
Matrix:
  • -0.24 0.52j -0.73 0.38j -0.80 -0.19j 0.36 0.45j
(1): Gate(Y, X, ...)
Name:
  • Gate(Y, X, ...)
Generators:
  • 'Y': 106.399
  • 'X': -187.142
  • 'Z': -128.224
Matrix:
  • -0.41 0.58j -0.13 0.69j 0.53 0.46j -0.68 -0.20j
 
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:
  • 1.00 1.00j 1.00j 1.00
 
 
Marker 0
Compilation tools may only recompile cycles with equal markers.
(0): Gate(Y, X, ...)
Name:
  • Gate(Y, X, ...)
Generators:
  • 'Y': 69.523
  • 'X': -94.594
  • 'Z': -243.065
Matrix:
  • -0.39 0.87j -0.07 0.30j 0.27 0.15j -0.90 -0.30j
(1): Gate(Y, X, ...)
Name:
  • Gate(Y, X, ...)
Generators:
  • 'Y': 164.075
  • 'X': 54.822
  • 'Z': 88.433
Matrix:
  • 0.12 -0.45j -0.59 -0.66j 0.87 0.17j -0.33 0.33j
 
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:
  • 1.00 1.00j 1.00j 1.00
 
 
Marker 0
Compilation tools may only recompile cycles with equal markers.
(0): Gate(Y, X, ...)
Name:
  • Gate(Y, X, ...)
Generators:
  • 'Y': 116.477
  • 'X': -3.951
  • 'Z': 6.082
Matrix:
  • 0.16 0.50j -0.22 -0.82j 0.16 0.83j 0.07 0.52j
(1): Gate(Y, X, ...)
Name:
  • Gate(Y, X, ...)
Generators:
  • 'Y': 64.58
  • 'X': -71.542
  • 'Z': 21.932
Matrix:
  • -0.04 0.67j -0.37 -0.64j -0.68 0.30j -0.36 0.57j
 
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:
  • 1.00 1.00j 1.00j 1.00
 
 
Marker 0
Compilation tools may only recompile cycles with equal markers.
(0): Gate(Y, X, ...)
Name:
  • Gate(Y, X, ...)
Generators:
  • 'Y': -6.86
  • 'X': 64.633
  • 'Z': 97.594
Matrix:
  • -0.78 -0.41j -0.47 0.02j -0.46 0.12j 0.63 -0.62j
(1): Gate(Y, X, ...)
Name:
  • Gate(Y, X, ...)
Generators:
  • 'Y': -89.612
  • 'X': -50.837
  • 'Z': 109.75
Matrix:
  • -0.69 -0.30j 0.36 -0.56j 0.29 0.59j 0.72 -0.21j

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

[5]:
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)
../../_images/guides_compilation_synthesis_8_0.png

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Download this file as Jupyter notebook: synthesis.ipynb.