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Noise tailoring for scalable quantum computation via randomized compiling
Joel J. Wallman, Joseph Emerson
TL;DR
General noise can be reduced to effective Pauli processes through randomized compiling. The technique reduces worst-case errors by orders of magnitude, supports threshold estimation from Pauli-noise estimates, and remains robust under some gate-dependent errors, while robustness to longer-timescale non-Markovian noise remains open.
Problem
The paper addresses how to reduce general, including partially coherent, noise into a form suitable for reliable quantum computation.
Method
Randomized compiling independently samples and compiles random twirling gates into sequential operations, tailoring noise into stochastic Pauli processes.
Results
The technique reduces worst-case errors by orders of magnitude and enables threshold estimates for general noise models to be obtained from Pauli-noise threshold estimates.
Takeaways & Limitations
The tailored channel's average error rate can be compared directly with fault-tolerance thresholds using efficient experiments such as randomized benchmarking.
Takeaways & Limitations
Robustness to noise that remains non-Markovian over timescales longer than a typical gate time is a significant open problem.
Abstract
from arXiv · showhide
Quantum computers are poised to radically outperform their classical counterparts by manipulating coherent quantum systems. A realistic quantum computer will experience errors due to the environment and imperfect control. When these errors are even partially coherent, they present a major obstacle to achieving robust computation. Here, we propose a method for introducing independent random single-qubit gates into the logical circuit in such a way that the effective logical circuit remains unchanged. We prove that this randomization tailors the noise into stochastic Pauli errors, leading to dramatic reductions in worst-case and cumulative error rates, while introducing little or no experimental overhead. Moreover we prove that our technique is robust to variation in the errors over the gate sets and numerically illustrate the dramatic reductions in worst-case error that are achievable. Given such tailored noise, gates with significantly lower fidelity are sufficient to achieve fault-tolerant quantum computation, and, importantly, the worst case error rate of the tailored noise can be directly and efficiently measured through randomized benchmarking experiments. Remarkably, our method enables the realization of fault-tolerant quantum computation under the error rates observed in recent experiments.
A. Standardized form for compiled quantum circuits
The paper organizes compiled circuits into cycles of easy single-qubit gates followed by hard gates, separating operations by their relative implementation noise. This standardized form supports universal and fault-tolerant circuit architectures.
- Compiled circuits are divided into easy and hard gate sets according to their relative implementation noise or physical realization.Easy gates are less noisy, while hard gates are more difficult to implement.
- The canonical choice uses Pauli-generated gates and the phase gate R as easy gates, with H, the π/8 gate, and controlled-Z as hard gates.
- These gate sets are universal and suit fault-tolerant settings including CSS codes, color codes, and the surface code.
- The bare circuit is organized into K cycles, each containing a round of easy gates followed by disjoint hard gates.The hard-gate round is represented by G_k, and the convention G_K = I makes the circuit end with easy gates.
- Randomized compiling inserts twirling gates around easy gates and compiles them into an equivalent circuit with the same number of elementary gates.The compiled randomized circuit remains logically equivalent to the bare circuit.
B. Randomized compiling
Randomized compiling independently samples twirling gates and compiles their corrections into adjacent easy gates, preserving the logical circuit while averaging noise toward stochastic channels. The protocol has modest classical recompilation overhead, and its general-noise guarantee remains incomplete.
- For qubits, the Pauli set consists of the four Hermitian, unitary operators {I, X, Y, Z}, while generalized Paulis provide the qudit construction.
- Each easy-gate round is replaced by randomized dressed gates using independently sampled twirling gates and correction operators that undo prior randomization.
- The dressed gates are compiled into single rounds of elementary gates rather than implemented as three separate rounds.The correction gates must belong to the easy-gate set for this compilation.
- Averaging over independently sampled twirling sequences produces the tailored stochastic noise channel, although each individual sequence need not realize that channel.Dressed gates can be recompiled in advance on a classical computer or applied on the fly with fast classical control.
- The scheme is expected to handle more general noise approximately, but a fully general proof remains an open problem.
C. Robustness to arbitrary independent errors on the hard gates
Under Markovian noise with gate-independent easy-gate errors, randomized compiling exactly tailors each nonfinal cycle into stochastic Pauli noise while permitting arbitrary hard-gate dependence. The final cycle and measurements require additional treatment.
- The hard-gate noise may depend arbitrarily on the implemented hard gates and may include cross-talk and multi-qubit correlations.
- Randomly sampled twirling gates tailor noise at every time step except the last into stochastic Pauli noise when easy-gate noise is gate-independent.
- Correction gates are chosen as inverses of the randomizing gates after commuting through the hard-gate round, enabling the cycle-wise noise averaging.
- Using the Pauli twirling set yields a Pauli channel, and the same averaged channel is obtained for any unitary 1-design.
- Randomizing the final Pauli frame and classically relabeling measurement outcomes can tailor noise in the final single-qubit Clifford-and-measurement round.
- The technique also applies to quantum nondemolition measurements on a subset of qubits by randomizing unmeasured qubits before and after measurement.
D. Robustness to independent errors on the easy gates
The technique remains effective with gate-dependent errors on easy gates: it adds a relatively small error, especially when the twirling group is normalized by the computational group, while easy-gate noise still limits usable circuit length.
- Residual control errors can create gate-dependent coherent errors on easy gates, but the technique is designed to retain noise-tailoring benefits in this setting.
- Gate-dependent noise on easy gates depends on previous twirling gates, preventing independent noise assignment to each tailored circuit cycle.
- Theorem 2 bounds the additional error from replacing gate-independent easy-gate noise with gate-dependent noise.
- When T is normalized by C, the additional error is relatively small and the stronger bound applies in many practical cases, including Pauli twirling with Pauli-and-R gates.
- Easy-gate noise limits either the useful length without error correction or the distance between error-correction rounds.
- Even without randomized compiling, Pauli noise requires r(Tk) ≪1/K because it accumulates linearly and must remain small for realistic circuits.
- The infidelity-based bound can be estimated efficiently through randomized benchmarking, although the authors expect it to be loose because it uses the triangle inequality.
E. Numerical simulations
Numerical simulations examine randomized compiling in six-qubit circuits, comparing bare and tailored implementations under gate-dependent over-rotation noise. Tailoring produces larger relative improvements at lower infidelity and reduces typical total error while both circuit types scale approximately linearly with circuit length.
- Simulation setup: The simulations use six-qubit circuits with Pauli and phase gates as easy gates, and Hadamard, T, and controlled-Z gates as hard gates.These circuits are universal and use the canonical easy–hard gate division.
- Figure 4: Approximately a factor of two on a log scale separates bare and tailored errors as infidelity decreases, indicating a larger relative improvement at lower infidelity.The reported separation mirrors the distinction between worst-case errors for stochastic and unitary channels, although these simulations do not maximize over preparations and measurements.
- Simulation setup: Figure 4 compares variational-distance errors for bare and tailored circuits, averaging each tailored result over 103 randomizations of the corresponding bare circuit.The error is evaluated using computational-basis measurement distributions rather than maximization over preparations and measurements.
- Figure 5: Both bare and tailored circuits show typical error growing approximately linearly with circuit length, while tailoring reduces the contribution from each error location.The simulations use controlled-Z infidelity 10^-3 and easy-gate infidelities 10^-5, with circuit length varying from five to one hundred alternating rounds.
II. DISCUSSION
The discussion presents randomized compiling as a way to reduce arbitrary Markovian noise to effective Pauli noise and extend the method across universal fault-tolerant gate sets. It reports orders-of-magnitude worst-case-error reductions and identifies robustness to long-lived non-Markovian noise as an open problem.
- Discussion: Arbitrary Markovian noise processes can be reduced to effective Pauli processes by compiling uniformly random gate sets into sequential operations.The average error rate of the tailored composite channel can be estimated using interleaved randomized benchmarking.
- Discussion: The technique applies directly to universal quantum-computing gate sets, including a large class of fault-tolerant proposals.It requires only local gates to tailor general multi-qubit noise into Pauli noise.
- Discussion: The numerical simulations demonstrate that the technique can reduce worst-case errors by orders of magnitude.The discussion further states that fault-tolerant scaling can exponentially improve the effect of a physical-level reduction.
- Open problem: Robustness to noise that remains non-Markovian longer than a typical gate time is identified as a significant open problem.The authors expect randomized gates may suppress such noise similarly to randomized dynamic decoupling, but present this as an expectation.
III. METHODS
The method analyzes tailored circuits under gate-dependent and gate-independent noise, using randomized twirling averages and diamond-norm bounds to quantify their difference and error contributions.
- Figure 5 shows approximately linear error growth with circuit length for both bare and tailored circuits, indicating suppression at individual error locations.
- The proof compares tailored circuits with gate-dependent and gate-independent noise on the easy gates.
- The analysis represents tailored circuits as non-commutative products whose expectation is taken over the twirling gates in each cycle.
- Averaging over twirling gates before applying the triangle inequality yields a substantially improved error bound.
- For local noise, the analysis treats the noise as a tensor product of single-qubit channels and averages each qubit’s noise over dressed gates.