Real Randomized Benchmarking

1 Introduction

• Quantum process tomography is impractical because it assumes lower-error state preparation and measurement and requires exponentially many configurations as qubit count grows.Compressed-sensing improvements do not remove the exponential growth described here [17] [26].Source: §1 · p. 1
• Randomized benchmarking estimates average fidelity between realized and ideal implementations of a gate group from survival-probability decay over random sequences of varying lengths; its effort scales efficiently with qubit count and it is robust against measurement and state-preparation errors.Source: §1 · p. 1
• Real RB uses gates from the real Clifford group, and its protocol estimates survival-probability decay over random sequences of varying lengths.The error channel is assumed gate- and time-independent; relaxing gate dependence is left for future work.Source: §1 · pp. 1–2
• The real Clifford group is an orthogonal 2-design, allowing efficient estimation of the average fidelity of its experimental implementation.Source: §1 · p. 2
• Real RB is motivated by codes that may lack all transversal Clifford gates and by the need to characterize fidelity over real-valued states relevant to rebit computation.The real Clifford group may be accessible in such codes; a variant was used inside the code [28].Source: §1 · p. 2
• The parameter b is linearly proportional to average rebit fidelity, and b together with c gives average fidelity, providing finer information about real Clifford gate implementations.The rebit average is taken with respect to the orthogonal group.Source: §1 · p. 3

2 Group twirl

• The group twirl is an orthogonal projection onto the commutant, and groups with the same commutant produce equivalent twirls.A design captures this equivalence and can be useful when the replacement group is smaller or simpler to implement.Source: §2 · p. 5
• A unitary t-design is defined by equality of commutants, and the paper extends this design idea to reference groups including the orthogonal group.The orthogonal t-design definition is given for a matrix group acting on C^d.Source: §2 · p. 5

2.1 Group twirl over the orthogonal group

• The orthogonal-group commutant is spanned by three projections corresponding to identity, antisymmetric, and traceless symmetric components.The corresponding normalized states form the convex hull of commutant density operators.Source: §2.1 · pp. 5–6
• The orthogonal twirl of a channel has components corresponding to the identity, Werner-Holevo, and completely depolarizing channels.The paper uses channel-state duality to identify these components.Source: §2.1 · p. 6
• The channel twirl and the corresponding twirl of its Jamiołkowski state are equivalent descriptions, and the orthogonal twirl gives an explicit channel representation.Source: §2.1 · p. 6

• The real Clifford group C(n) is the normalizer in O(2^n) of the real Pauli group E(n).E(n) is generated by tensor products of X and Z with 2×2 identity matrices.Source: §3 · pp. 6–7
• The real Clifford group C(n) is an orthogonal 2-design, the main mathematical ingredient for real randomized benchmarking.The proof uses its inclusion in the real orthogonal group and equality of their commutants.Source: §3 · p. 7
• The real Clifford group is an orthogonal 3-design but not an orthogonal 4-design.Source: §3 · p. 7

• The complex Clifford group is a unitary 2-design and 3-design, but not an exact unitary 4-design; the real Clifford group shares these stated design properties.Source: §4 · p. 8
• The real Clifford group is a subgroup of the complex Clifford group, and the text connects their t-design properties for t = 2 and 3 to invariant harmonic polynomials.Taking real and imaginary parts of invariant polynomials can produce zero polynomials, so the absence of real invariants does not establish the absence of complex ones.Source: §4 · pp. 8–9
• The real Clifford group is an algebraic almost unitary 2-design as well as an exact orthogonal 2-design.The Clifford groups are also described as algebraic almost 4-designs, with one additional commutant generator relative to the unitary group.Source: §4 · p. 9
• Additional commutant factors yield additional fidelity-decay terms, allowing finer-grained error estimates, although fitting becomes poorly conditioned for large l.For the real Clifford group, there is only one extra decay term, and the text says efficient measurement-based data processing can fit the model.Source: §4 · p. 9
• The section points to an efficient method for obtaining a Haar sample from the real Clifford group as an ingredient of the real randomized-benchmarking protocol.Source: §4 · p. 9

5.1 Structure of the real Clifford group, and orthogonal transformations

• A symplectic matrix represents an element of O+(2n, 2) exactly when its columns are singular and form a hyperbolic basis.The quadratic form Q satisfies Q(x + x′) − Q(x) − Q(x′) = [x, x′].Source: §5.1 · pp. 9–10
• To sample the real Clifford group, draw an element of O+(2n, 2), implement its symplectic matrix as a Clifford operation, then multiply by a random element of E(n).Clifford representatives differing by Pauli actions induce the same symplectic matrix; the real-valued representative corresponds to an element of O+(2n, 2).Source: §5.1 · p. 10

5.2 Efficient sampling from O+(2n, 2)

• The sampling procedure constructs hyperbolic pairs iteratively and returns a matrix whose columns form a hyperbolic basis.At each iteration it selects a nonzero singular vector, then a vector pairing to one with it, and extends the basis.Source: §5.2 · pp. 10–11
• Every hyperbolic basis is equally likely under the procedure, so it samples uniformly from O+(2n, 2).Source: §5.2 · p. 11

• The physical quantum channel is written as the ideal channel composed with an error channel, and its quality is assessed using average fidelity and average rebit fidelity.Source: §6 · pp. 11–12
• Average rebit fidelity is averaged with respect to the orthogonal group, and the two fidelity quantities relate to parameters that randomized benchmarking can estimate.Source: §6 · p. 12
• For real density matrices, the twirled channel acts as a depolarizing channel with parameter b, making the analysis relevant to rebit computations and stabilizer circuits.The real Clifford group plays the role of the complex Clifford group for stabilizer circuits on rebits.Source: §6 · p. 12

7 Real randomized benchmarking protocol

• In the real randomized benchmarking protocol, quantum gates are taken from the real Clifford group, and the error quantum channel is assumed to be both gate- and time-independent.Source: §7 · p. 12
• Each sequence uses m independent random real Clifford gates and a final gate chosen so the ideal net operation is the identity.The protocol then measures survival probability and averages over M random realizations at each sequence length.Source: §7 · p. 13
• Averaging repeated sequences approximates the average sequence fidelity, with A, B, and C set by preparation and measurement and b and c by noise.The protocol obtains the decay curve by varying sequence length m.Source: §7 · pp. 12–13
• The final inverse gate can be found efficiently using the Gottesman–Knill theorem.The paper recommends exponentially spaced sequence lengths from 4 to around 1/(1 − F̄_R) to avoid gate-dependent effects and optimize information gain.Source: §7 · pp. 13–14

8 Fidelity decay and parameter estimation

• The expected sequence fidelity follows A + b^mB + c^mC, with A, B, and C depending only on preparation and measurement and b and c on average noise.This form follows from averaging over the real Clifford group, which is an orthogonal 2-design.Source: §8 · pp. 14–15
• With a = 1, the averaged-fidelity model applies to any fixed prepared state and fixed POVM measurement.A, B, and C are independent of the noise channel.Source: §8 · p. 16
• Directly fitting the multi-exponential decay can be poorly conditioned, while the differencing approach reduces the fit to simpler decay curves.The paper also assumes preparation and measurement noise is independent of the particular state or measurement, noting this will not hold exactly in practice.Source: §8 · pp. 16–17
• Differences between data sets yield separate decay curves ΔB b^m and ΔC c^m, isolating b and c.The data sets vary symmetric or antisymmetric Pauli eigenstate preparations and the corresponding ±1 eigenstate measurement projectors.Source: §8 · pp. 16–17
• Fitting the differenced data to ΔB b^m or ΔC c^m gives separate estimates without strong covariance between b and c.The fits are simpler exponential models; imperfect cancellation is expected in practice, but the transformation still greatly enhances contrast.Source: §8 · p. 17

Appendix

• The appendix presents the frame-potential route as an alternative way to establish that the real Clifford group is an orthogonal 2-design.It follows the approach of [25].Source: Appendix · p. 22
• A set of real orthogonal matrices is an orthogonal 2-design if and only if its frame potential equals 3.The appendix also states that this is equivalent to the associated symmetry having no more than three irreducible representations.Source: Appendix · pp. 22–24
• The proof relates the frame potential to the character of the associated representation, whose irreducible components determine its inner product.The character decomposes into a sum of irreducible characters, and the frame-potential criterion is met when the associated representation has exactly three irreducible components.Source: Appendix · pp. 24–25
• The real Clifford group is an orthogonal 2-design.The proof uses the result that its associated representation has three irreducible representations.Source: Appendix · p. 25