SolovayKitaev

class qiskit.transpiler.passes.SolovayKitaev(*args, **kwargs)

Bases: TransformationPass

Approximately decompose 1q gates to a discrete basis using the Solovay-Kitaev algorithm.

The Solovay-Kitaev theorem [1] states that any single qubit gate can be approximated to arbitrary precision by a set of fixed single-qubit gates, if the set generates a dense subset in $SU(2)$. This is an important result, since it means that any single-qubit gate can be expressed in terms of a discrete, universal gate set that we know how to implement fault-tolerantly. Therefore, the Solovay-Kitaev algorithm allows us to take any non-fault tolerant circuit and rephrase it in a fault-tolerant manner.

This implementation of the Solovay-Kitaev algorithm is based on [2].

For example, the following circuit

     ┌─────────┐
q_0: ┤ RX(0.8) ├
     └─────────┘

can be decomposed into

global phase: 7π/8
     ┌───┐┌───┐┌───┐
q_0: ┤ H ├┤ T ├┤ H ├
     └───┘└───┘└───┘

with an L2-error of approximately 0.01.

Examples

Per default, the basis gate set is ["t", "tdg", "h"]:

import numpy as np
from qiskit.circuit import QuantumCircuit
from qiskit.transpiler.passes.synthesis import SolovayKitaev
from qiskit.quantum_info import Operator
 
circuit = QuantumCircuit(1)
circuit.rx(0.8, 0)
 
print("Original circuit:")
print(circuit.draw())
 
skd = SolovayKitaev(recursion_degree=2)
 
discretized = skd(circuit)
 
print("Discretized circuit:")
print(discretized.draw())
 
print("Error:", np.linalg.norm(Operator(circuit).data - Operator(discretized).data))

Original circuit:
   ┌─────────┐
q: ┤ Rx(0.8) ├
   └─────────┘
Discretized circuit:
global phase: 7π/8
   ┌───┐┌───┐┌───┐
q: ┤ H ├┤ T ├┤ H ├
   └───┘└───┘└───┘
Error: 2.828408279166474

For individual basis gate sets, the generate_basic_approximations function can be used:

from qiskit.synthesis import generate_basic_approximations
from qiskit.transpiler.passes import SolovayKitaev
 
basis = ["s", "sdg", "t", "tdg", "z", "h"]
approx = generate_basic_approximations(basis, depth=3)
 
skd = SolovayKitaev(recursion_degree=2, basic_approximations=approx)

References

[1]: Kitaev, A Yu (1997). Quantum computations: algorithms and error correction.

Russian Mathematical Surveys. 52 (6): 1191–1249. Online.

[2]: Dawson, Christopher M.; Nielsen, Michael A. (2005) The Solovay-Kitaev Algorithm.

arXiv:quant-ph/0505030.

Parameters

• recursion_degree – The recursion depth for the Solovay-Kitaev algorithm. A larger recursion depth increases the accuracy and length of the decomposition.
• basic_approximations – The basic approximations for the finding the best discrete decomposition at the root of the recursion. If a string, it specifies the .npy file to load the approximations from. If a dictionary, it contains {label: SO(3)-matrix} pairs. If None, a default based on the H, T and Tdg gates up to combinations of depth 10 is generated.

Attributes

is_analysis_pass

is_transformation_pass

Methods

execute

name

run

update_status