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Scalable randomized benchmarking of non-Clifford gates

Andrew W. Cross, Easwar Magesan, Lev S. Bishop, John A. Smolin, Jay M. Gambetta

arXiv:1510.02720v1quant-ph

TL;DR

The paper develops orbit-based twirling analyses for Pauli channels under cX and cZ operations. These reductions describe the channels with five, four, and ultimately three parameters after successive twirls.

Abstract

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Randomized benchmarking is a widely used experimental technique to characterize the average error of quantum operations. Benchmarking procedures that scale to enable characterization of $n$-qubit circuits rely on efficient procedures for manipulating those circuits and, as such, have been limited to subgroups of the Clifford group. However, universal quantum computers require additional, non-Clifford gates to approximate arbitrary unitary transformations. We define a scalable randomized benchmarking procedure over $n$-qubit unitary matrices that correspond to protected non-Clifford gates for a class of stabilizer codes. We present efficient methods for representing and composing group elements, sampling them uniformly, and synthesizing corresponding $\mathrm{poly}(n)$-sized circuits. The procedure provides experimental access to two independent parameters that together characterize the average gate fidelity of a group element.

Supplemental Materials: Scalable randomized benchmarking of non-Clifford gates

The supplementary material establishes the algebraic structure of Gm, derives efficient representation and sampling procedures, and develops the twirling analysis used for scalable benchmarking.

Supplemental Materials: Scalable randomized benchmarking of non-Clifford gates

  • Gm decomposes into diagonal phase operators and permutation matrices, enabling canonical rewriting of each element into structured components.The permutation subgroup is generated by bitflips and CNOT gates, while diagonal elements are represented through polynomials over Zm.Source: Supplemental Materials: Scalable… · p. 6
  • Every monomial xα in Φ−1(W(n)) has a coefficient that is a multiple of 2|α|−1.Source: Supplemental Materials: Scalable… · p. 7
  • The group order grows rapidly with n but remains comparable to the Clifford group for the explicitly constructed small groups.Table I reports group orders for non-Clifford matrix groups with n ≤3.Source: Supplemental Materials: Scalable… · p. 7, Table I · p. 7
  • Uniform sampling combines random affine transformations with independently sampled polynomial coefficients using polynomially bounded randomness in the scalable case.The matrix component is sampled from GLn(F2), while O(nk) coefficients determine the diagonal polynomial component.Source: Supplemental Materials: Scalable… · p. 7
  • The twirling analysis reduces Pauli-channel parameters through orbit averaging, with the cX-twirl producing five parameters and the cZ-twirl reducing them to four.The cZ orbit structure averages parameters associated with X- and Y Y-type Pauli operators.Source: Supplemental Materials: Scalable… · p. 8

OZ(1) = Z \ {I}

The cX and cZ actions partition Pauli operators into explicitly characterized orbit families based on supports, parity, and representative strings.

OZ(1) = Z \ {I}

  • The cX orbits distinguish Pauli operators by support and the parity of Y-type factors, preserving the separation between X-type and Z-type factors.The listed OY(1) and OY(1,2) families use odd and even strings on the support.Source: OZ(1) = Z \ {I} · p. 8

OI = {I}

The later twirls average non-Z Pauli sectors and exploit diagonal transfer-matrix structure, reducing the channel to three parameter blocks.

OI = {I}

  • The cZ-twirl combines X(u) and X(u)Z(u*) orbit classes, averaging their parameters into βX−Y Y and leaving four channel parameters.The averaging is uniform because the relevant orbits have equal size.Source: OI = {I} · p. 8
  • The diagonal phase twirl makes X and Y tensor factors contribute equally while Z factors remain unchanged, reducing the Pauli channel to three parameters.The resulting parameter βR averages βX−Y Y and βY.Source: OI = {I} · pp. 8–9
  • Diagonal gates produce block-diagonal transfer matrices with identity Z blocks and zero off-diagonal blocks, constraining the final twirled map.The inverse relation R(U†) = R(U)^−1 is used for the final twirl.Source: OI = {I} · p. 9
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