Quantum-Twirling

A Python package designed for fast implementation of Pauli Twirling in Qiskit circuits.

Features

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• It includes a fast built-in dictionary of Pauli Twirling Sets for Qiskit two-qubit Clifford gates CX, CY, CZ, CH, DCX, CSX, CSdg, ECR, SWAP, iSWAP.
• It can generate Pauli Twirling Sets for any given two-qubit gate.
• If used with an instance of Qiskit's PassManager, it can implement randomized Pauli Twirls to a circuit containing two-qubit gates.

Contents

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• Features
• What is Pauli Twirling?
  • Implementation of Pauli Twirling in Academic Literature
• Function Documentation
• Usage and Examples
  • Load
  • Generate
  • Twirl Qiskit Circuits
• Acknowledgements

What is Pauli Twirling?

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Pauli Twirling is a powerful error supression/mitigation technique for quantum circuits. By combining the results from many random, yet logically equivalent circuits, it converts an arbitrary noise channel into a stochastic Pauli error channel, thus suppressing off-diagonal coherent error contributions. It is implemented by sandwiching a two-qubit Clifford gate between randomly sampled single-qubit twirling gates drawn from a set of Pauli operations.

Crucially, the gates are selected such that the net operation, in the absence of noise, is unchanged. That is, the Pauli Twirling Set for a two-qubit Clifford gate is a set of four Pauli gates that when pre-pended and appended to a circuit containing the two-qubit gate, the unitary is equal to that of the single two-qubit gate alone.

Implementation of Pauli Twirling in Academic Literature

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• Purification of noisy entanglement and faithful teleportation via noisy channels (Bennet, et. al., 1996)
• Mixed state entanglement and quantum error correction (Bennet, et. al., 1996)
• Practical Implementations of Twirl Operations (Anwar, Short, and Jones, 2008)
• Scalable error mitigation for noisy quantum circuites produces competitive expectation values (Kim, et. al., 2021)
• Constructing smaller Pauli twirling sets for arbitrary error channels (Cai and Benjamin, 2018)

Dictionary-Supported Qiskit Gates

Native Pauli Twirling Sets are included for the following common two-qubit Clifford gates in Qiskit:

• CXGate (Controlled X Gate, a.k.a. CNOT Gate)
• CYGate (Controlled Y Gate)
• CZGate (Controlled Z Gate)
• CHGate (Controlled Hadamard Gate)
• DCXGate (Double Controlled X Gate, a.k.a., Double CNOT Gate)
• CSXGate (Controlled Square Root of X Gate)
• CSdgGate (Controlled S^dagger Gate)
• ECRGate (Echoed Cross-Resonance Gate)
• SWAPGate (Swap Gate)
• iSWAPGate (Imaginary Swap Gate)

Documentation

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Module: load_pauli_twirling_sets

Import Module:

from quantum_twirling import load_pauli_twirling_sets

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Load the entire Pauli Twirling Sets dictionary:

quantum_twirling.load_pauli_twirling_sets.load_pauli_twirling_dict()

Output is the full dictionary. Each gate's Pauli Twirling Sets can be accessed by passing the lowercase name of the gate to the dictionary, e.g: cx_twirls = pauli_twirling_dict['cx']

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Return list of CX Gate Pauli Twirling Sets:

quantum_twirling.load_pauli_twirling_sets.load_cxgate_twirls()

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Return list of CY Gate Pauli Twirling Sets:

quantum_twirling.load_pauli_twirling_sets.load_cygate_twirls()

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Return list of CZ Gate Pauli Twirling Sets:

quantum_twirling.load_pauli_twirling_sets.load_czgate_twirls()

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Return list of CH Gate Pauli Twirling Sets:

quantum_twirling.load_pauli_twirling_sets.load_chgate_twirls()

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Return list of DCX Gate Pauli Twirling Sets:

quantum_twirling.load_pauli_twirling_sets.load_dcxgate_twirls()

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Return list of CSX Gate Pauli Twirling Sets:

quantum_twirling.load_pauli_twirling_sets.load_csxgate_twirls()

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Return list of CSdg Gate Pauli Twirling Sets:

load_pauli_twirling_sets.load_csdggate_twirls()

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Return list of ECR Gate Pauli Twirling Sets:

load_pauli_twirling_sets.load_ecrgate_twirls()

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Return list of SWAP Gate Pauli Twirling Sets:

load_pauli_twirling_sets.load_swapgate_twirls()

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Return list of iSWAP Gate Pauli Twirling Sets:

load_pauli_twirling_sets.load_iswapgate_twirls()

Module: generate_pauli_twirling_sets

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Import Module:

from quantum_twirling import generate_pauli_twirling_sets

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generate_pauli_twirling_sets(TwoQubitGate())

• The TwoQubitGate() should to be a two-qubit instance of the Qiskit Gate class.
• Finds Pauli Twirls for the given Gate; returns a list of the form: [(gates,phase), (gates,phase), ...)].

Module: pauli_twirling

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Import Module:

from quantum_twirling import pauli_twirling

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CLASS PauliTwirling(PassManager)

• Bases: PassManager (PassManager Class in Qiskit) To twirl all instances of a given two-qubit gate in a quantum circuit, attach PauliTwirling pass to a Qiskit PassManager. Tell the PauliTwirling pass to twirl gates of a certain type by providing a list of Pauli Twirling Sets for a given two-qubit gate. Seed the pass; Pauli Twirls are randomly applied to circuit.

pm = PassManager([PauliTwirling(gate_twirling_sets, seed=" ")])

To use the pass, call pm.run(qc) where qc is a quantum circuit. All gates of type given in PassManager will be randomly Pauli Twirled. Subsequent calls of pm.run(qc) will yield other circuits that are likewise randomly twirled.

Usage and Examples

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import qiskit
import quantum_twirling
from quantum_twirling import load_pauli_twirling_sets
from quantum_twirling.generate_pauli_twirling_sets import generate_pauli_twirling_sets
from quantum_twirling.pauli_twirling import PauliTwirling

Load Pauli Twirling Sets

Load the entire existing dictionary with load_pauli_twirling_dict():

twirl_dict = load_pauli_twirling_sets.load_pauli_twirling_dict()

Load individual gate twirling sets with: cxgate_twirls(), etc:

cx_twirls = load_pauli_twirling_sets.load_cxgate_twirls()
cy_twirls = load_pauli_twirling_sets.load_cygate_twirls()
cz_twirls = load_pauli_twirling_sets.load_czgate_twirls()
ch_twirls = load_pauli_twirling_sets.load_chgate_twirls()
dcx_twirls = load_pauli_twirling_sets.load_dcxgate_twirls()
csx_twirls = load_pauli_twirling_sets.load_csxgate_twirls()
csdg_twirls = load_pauli_twirling_sets.load_csdggate_twirls()
ecr_twirls = load_pauli_twirling_sets.load_ecrgate_twirls()
swap_twirls = load_pauli_twirling_sets.load_swapgate_twirls()
iswap_twirls = load_pauli_twirling_sets.load_iswapgate_twirls()

Generate Pauli Twirling Sets

from quantum_twirling import generate_pauli_twirling_sets
twirling_sets = generate_pauli_twirling_sets(TwoQubitGate)

Twirl Qiskit Circuits

Import the PauliTwirling class:

from quantum_twirling.pauli_twirling import PauliTwirling

Create a two-qubit quantum circuit to Pauli Twirl. Note that the two-qubit gate in this circuit is a CX gate. Thus, we can use our cx_dict retrieved from get_pauli_twirling_dict('cx'), earlier.

qc = QuantumCircuit(3)
qc.h(1)
qc.cx(1,0)
qc.cx(1,2)
qc.draw('mpl')

To use the pass, attach to a PassManager. Here, we do that and tell our PauliTwirling pass to twirl CX gates in our cx_dict.

pm = PassManager([PauliTwirling('cx_dict', seed=54321)])

Call pm.run once and draw the circuit:

twirl_qc = pm.run(qc)
twirl_qc.draw('mpl')

Subsequent calls will yield other circuits that are likewise randomly twirled.

Acknowledgements

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IBM Quantum: Generating Pauli-twirled circuits in Qiskit