Add the Solovay-Kitaev algorithm to the transpiler passes

[LNoorl]

Summary

Add the Solovay-Kitaev algorithm from https://arxiv.org/abs/quant-ph/0505030 by C. Dawson and M. Nielsen to the transpiler passes.

Details and comments

For example, the following circuit

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

can be decomposed into

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

with an L2-error of approximately 0.01.

The code to do the above is:

import numpy as np
from qiskit.circuit import QuantumCircuit
from qiskit.circuit.library import TGate, HGate, TdgGate
from qiskit.converters import circuit_to_dag, dag_to_circuit
from qiskit.transpiler.passes import SolovayKitaevDecomposition
from qiskit.quantum_info import Operator

circuit = QuantumCircuit(1)
circuit.rx(0.8, 0)
dag = circuit_to_dag(circuit)

print('Original circuit:')
print(circuit.draw())

basis_gates = [TGate(), TdgGate(), HGate()]
skd = SolovayKitaevDecomposition(recursion_degree=2, basis_gates=basis_gates)

discretized = dag_to_circuit(skd.run(dag))

print('Discretized circuit:')
print(discretized.draw())

print('Error:', np.linalg.norm(Operator(circuit).data - Operator(discretized).data))

Preliminary results

Here are some benchmark results for the QFT circuit on three qubits using the basis gate set h t tdg and combinations thereof of depth up to 10.

Error (in diamond norm):

Runtime:

TODO

• Fix the global phase
• Fix the accumulation roundoff errors
• Add test for multiqubit circuits
• Fix the basis gates in the unit tests
• Allow setting the basis gates by string
• Fix pylint

[Cryoris]

@mtreinish @alexanderivrii all set! Let me know if there's other changes we should add 🙂

[alexanderivrii]

@Cryoris, thank you for following up on the code reorganization comment. From this point of view, everything looks great, but I haven't really looked at anything else. My understanding was that @mtreinish is doing the actual review. (Of course, I would be happy to help, if required).

[mtreinish]

I think this is getting close, it'll be really good to get this functionality included. I left a few comments and questions inline, the biggest thing still to do from my PoV are fix the docs so we're actually including them in the build.

[mtreinish]

Is there a reason to just do GateSequence([gate.inverse() for gate in reversed(self.gates)])? I think this question holds for a bunch of places we're constructing new GateSequence objects.

[Cryoris]

Yeah because we can more efficiently compute the values that would otherwise be computed from scratch in the __init__ 🙂 I'll leave a comment in the code to make that clear

[mtreinish]

We should add thew transpiler pass to this list and re-export it from this module so people can do from qiskit.transpiler.passes import SolovayKitaev

[mtreinish]

Is there a reason to keep the save functionality around since the computation is much faster now?

[Cryoris]

Yeah, it's likely still slow for large basis sets/depths or you'd quickly like to restart a calculation 🙂

[mtreinish]

This LGTM now thanks for the updates. I pushed a couple commits to fix the docs build and clean things up a bit.