Source-linked AI summary
Randomized Benchmarking with Restricted Gate Sets
Winton G. Brown, Bryan Eastin
TL;DR
Standard randomized benchmarking requires sampling from a unitary 2 design, which can be challenging for restricted high-fidelity logical gate sets. The paper develops an analysis for randomized benchmarking using subgroups of the Clifford group that are not unitary 2 designs. Upper-bound estimates overstate entanglement infidelity by at most (2n + 2)/2n (a factor of two or less), and overestimate p by at most a factor of 2 independent of qubit count.
Problem
Standard randomized benchmarking requires sampling from a unitary 2 design, which can be challenging for restricted high-fidelity logical gate sets.
Source: §I · p. 1Method
The paper develops an analysis for randomized benchmarking using subgroups of the Clifford group that are not unitary 2 designs.
Source: §IV · p. 5Results
Upper-bound estimates overstate entanglement infidelity by at most (2n + 2)/2n (a factor of two or less), and overestimate p by at most a factor of 2 independent of qubit count.
Source: A. The Real Clifford Group · p. 4, B. Controlled-NOT and Pauli Gates · p. 5Takeaways & Limitations
The two studied subgroup protocols show that sampling a unitary 2 design is not required to estimate error probability conservatively within a factor of two or less.
Source: A. The Real Clifford Group · p. 4, B. Controlled-NOT and Pauli Gates · p. 5Preparing eigenstates from each Pauli block can be impractical when the gates needed to convert between blocks are impractical.
Source: §III · p. 3
Abstract
from arXiv · showhide
Standard randomized benchmarking protocols entail sampling from a unitary 2 design, which is not always practical. In this article we examine randomized benchmarking protocols based on subgroups of the Clifford group that are not unitary 2 designs. We introduce a general method for analyzing such protocols and subsequently apply it to two subgroups, the group generated by controlled-NOT, Hadamard, and Pauli gates and that generated by only controlled-NOT and Pauli gates. In both cases the error probability can be estimated to within a factor of two or less where the factor can be arranged to be conservative and to decay exponentially in the number of qubits. For randomized benchmarking of logical qubits even better accuracy will typically be obtained. Thus, we show that sampling a distribution which is close to a unitary 2 design, although sufficient, is not necessary for randomized benchmarking to high accuracy.
I. INTRODUCTION
Standard randomized benchmarking relies on sampling from unitary 2 designs, which can be difficult, especially for logical qubits with restricted high-fidelity gate sets. The paper asks whether some Clifford gates are truly necessary and says that in many cases they are not.
I. INTRODUCTION
- Standard protocols require sampling from a unitary 2 design, which can be challenging when the available high-fidelity logical gates are restricted.The paper examines whether the gates that create this difficulty are necessary for randomized benchmarking.Source: §I · p. 1
- Logical Clifford gates requiring state injection or distillation carry higher overhead and error, making them costly for near-term logical-qubit benchmarking.Including low-utility, poorly performing gates can also produce an overly pessimistic assessment; interleaved benchmarking may perform poorly when gate errors vary widely [13] [14].Source: §I · p. 1
- The article introduces a general analysis of randomized benchmarking with Clifford subgroups and applies it to the cases of interest.The introduction identifies Section III as the subgroup analysis and Section IV as the conclusion.Source: §I · p. 1
A. Setting and Assumptions
The analysis targets average fidelity of a group’s error channels and assumes errors can be modeled as completely positive maps with weak gate dependence. For simplicity, it further assumes the error channel is independent of the applied gate.
A. Setting and Assumptions
- The randomized-benchmarking goal is to determine the average fidelity of the error channels for a gate group.The protocol’s accuracy depends on properties of those channels.Source: A. Setting and Assumptions · p. 1
- For each target gate, the device evolution is modeled as a completely positive error map followed by the ideal unitary.This is the stated Markovianity condition.Source: A. Setting and Assumptions · p. 1
- Weak gate dependence requires that the error channel not depend strongly on which gate is applied; the norm used is the induced operator 1-norm.The displayed inequality defining the condition is not included in the supplied passage.Source: A. Setting and Assumptions · p. 1
- For simplicity, the analysis assumes the error channel is completely independent of the applied gate.The authors note prior complications with weak gate dependence [15] [16].Source: A. Setting and Assumptions · p. 1
B. Twirling
Twirling symmetrizes an error channel by averaging its conjugation over a gate group, reducing it to a form described by the group’s invariants. A unitary 2 design reproduces the relevant second-order averages of SU(2^n), but twirling need not use the full unitary group.
B. Twirling
- G-twirling averages a channel over random group gates applied at its input and inverse gates at its output, projecting the channel onto the group’s invariants.The resulting channel is invariant under conjugation by elements of G.Source: B. Twirling · p. 2
- Twirling by SU(2^n) has two linearly independent invariants: the identity channel and the completely depolarizing channel.The supplied passage identifies these as the two invariant channels.Source: B. Twirling · p. 2
- Under SU(2^n) twirling, every non-identity Pauli operator has the same channel eigenvalue, so the twirled channel is specified by one real number.Standard randomized benchmarking uses 2-design twirling, including with the Clifford group and certain subgroups.Source: B. Twirling · p. 2
- A group is a unitary 2 design when the specified second-order homogeneous polynomial averages match those from SU(2^n).Equivalently, the condition concerns the averages for non-identity Pauli operators; the supplied definition states that these groups suffice for twirling.Source: B. Twirling · p. 2
C. Standard Randomized Benchmarking Protocol
Standard randomized benchmarking estimates an error parameter from sequence fidelities measured at different sequence lengths. It uses random sequences from a unitary 2 design and a final inverse gate, then fits the length-dependent average fidelity.
C. Standard Randomized Benchmarking Protocol
- The target quantity is entanglement infidelity, or a simple function of it, rather than necessarily the average infidelity often quoted in randomized benchmarking.The paper focuses on entanglement infidelity because it is more relevant for multi-qubit states and is commonly used in quantum-error-correction simulations.Source: C. Standard Randomized Benchmarking… · p. 2
- The protocol prepares an initial state, applies a random sequence of gates drawn independently from a unitary 2 design, and appends a gate that makes the ideal sequence equal the identity.The experiment is repeated across sequence lengths to obtain average sequence fidelity.Source: C. Standard Randomized Benchmarking… · pp. 2–3
- The average fidelity across sequences of length l is fit to a decay curve to estimate the channel parameter λ.The measured fidelity also depends on state-preparation and measurement errors, which varying l helps isolate.Source: C. Standard Randomized Benchmarking… · p. 3
- Repeating each random sequence many times can reveal additional information about error-channel coherence and non-unitality [9].Using a different random sequence for each run minimizes the total number of measurements, but sequences are typically repeated for practical reasons.Source: C. Standard Randomized Benchmarking… · p. 3
III. CLIFFORD SUBGROUP TWIRLING
The paper adapts randomized benchmarking to Clifford subgroups that contain the Pauli group but are not unitary 2 designs. Subgroup twirling produces block-structured channels, so decay measurements can estimate error parameters, though preparing states across blocks may be impractical.
III. CLIFFORD SUBGROUP TWIRLING
- Pauli-group twirling converts an arbitrary channel into a stochastic Pauli channel, whose Pauli operators are eigenoperators.The paper uses decay of an eigenstate’s projector to estimate the corresponding error-channel fidelity.Source: §III · pp. 3–4
- Subgroup twirling partitions Pauli operators into orbits, producing a channel with one block-associated eigenvalue for each orbit block.The decay curve under imperfect mixing therefore has multiple components.Source: §III · p. 3
- Preparing and measuring an eigenstate from every block can be impractical when the gates needed to convert between blocks are impractical.The paper also describes extracting multiple decay curves from individual-qubit measurement results, or focusing on one curve of interest.Source: §III · p. 3
A. The Real Clifford Group
For the real Clifford group, twirling separates non-identity Pauli operators into real and imaginary blocks. Measuring the real-block decay gives a conservative estimate of entanglement infidelity, while logical-qubit error structure can make the estimate more accurate.
A. The Real Clifford Group
- Real-Clifford twirling creates two non-trivial blocks: non-identity Pauli operators with even versus odd numbers of Y elements.These are respectively the real and imaginary Pauli operators.Source: A. The Real Clifford Group · p. 4
- Estimating both block probabilities requires preparation and measurement in real and imaginary Pauli eigenspaces, which can be difficult when only real Clifford gates are available.In particular, codes with only transversal real Clifford gates may lack a straightforward logical Y-basis procedure; measuring only real eigenstates yields λ1.Source: A. The Real Clifford Group · p. 4
- Assuming p2 = 0, using the upper bound as the estimate of p overestimates entanglement infidelity by at most (2^n + 2)/2^n, a factor of two or less.For benchmarking logical qubits, the estimate is typically better because logical Y errors are strongly suppressed in many popular codes.Source: A. The Real Clifford Group · p. 4
B. Controlled-NOT and Pauli Gates
For the subgroup generated by controlled-NOT and Pauli gates, twirling yields four Pauli blocks. The paper bounds entanglement infidelity using decay constants accessible through X- and Z-basis preparations, with alternative bounds depending on which constants are measured.
B. Controlled-NOT and Pauli Gates
- Controlled-NOT action partitions Pauli operators into four blocks: Z-only, X-only, remaining non-identity even-Y, and odd-Y operators.The block eigenvalues characterize the twirled channel.Source: B. Controlled-NOT and Pauli Gates · p. 5
- With λ1 and λ2, entanglement infidelity can be bounded using eigenstates from B1 and B2, avoiding the often problematic logical Y basis.For logical qubits, X and Z preparation and measurement are often relatively straightforward in CSS codes.Source: B. Controlled-NOT and Pauli Gates · p. 5
- The upper-bound estimate overstates p by at most a factor of 2, independent of qubit count.This assumes p3 and p4 are zero; for logical qubits these probabilities are likely much smaller than p1 and p2.Source: B. Controlled-NOT and Pauli Gates · p. 5
- For n > 2, measuring a B3 Pauli eigenstate can extract λ3 and provide an alternative bound on p.The upper bound assumes p3 is the only nonzero probability and overestimates p by at most (2^n − 2)/(2^n − 4); the lower bound typically estimates logical-qubit infidelity better.Source: B. Controlled-NOT and Pauli Gates · p. 5
IV. CONCLUSION
The paper develops an analysis for randomized benchmarking with Clifford subgroups that are not unitary 2 designs and applies it to two gate sets. Its bounds show that high-accuracy approximate benchmarking is possible without sampling from a unitary 2 design.
IV. CONCLUSION
- For real Clifford gates, entanglement infidelity is estimated within a factor of (2^n + 2)/2^n; for controlled-NOT and Pauli gates, the estimate is within a factor of 2 or (2^n − 2)/(2^n − 4), depending on extracted decay constants.The latter result assumes the ability to prepare and measure both |0⟩ and |+⟩.Source: §IV · p. 5
- The method analyzes randomized benchmarking using Clifford subgroups that do not form unitary 2 designs.The paper applies it to the real Clifford group and the subgroup generated by controlled-NOT and Pauli gates.Source: §IV · p. 5