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Stability Experiments: The Overlooked Dual of Memory Experiments
Craig Gidney
TL;DR
Moving logical observables requires reliably combining many stabilizer measurements, yet no established experiment directly benchmarks that ability. This paper proposes stability experiments, which check a known stabilizer product across space. Simulations show these experiments can be forgiving at small patch sizes, while their scope is narrower than protection against all error mechanisms in full computations.
Problem
Moving logical observables can require multiplying hundreds or millions of stabilizers, but there is no established experiment for directly verifying reliable computation of large stabilizer products.
Source: §1 · p. 2Method
Results
Simulations show stability experiments have more forgiving finite-size thresholds than memory experiments; 4x4 stability experiments can suppress logical errors at physical noise rates roughly twice those needed for two-round memory experiments.
Source: §3 · pp. 5–6Takeaways & Limitations
Stability experiments can provide experimental access to high code distances at small patch sizes and quantify measurement rounds needed when moving a logical qubit.
Source: §4 · p. 8Stability experiments increase protection along only one dimension; protecting both observables and longer travel distances requires widening and lengthening the patch, which makes the experiment harder.
Source: §4 · p. 8
Abstract
from arXiv · showhide
Topological quantum computations are built on a foundation of two basic tasks: preserving logical observables through time and moving logical observables through space. Memory experiments, which check how well logical observables are preserved through time, are a well established benchmark. Strangely, there is no corresponding well established benchmark for moving logical observables through space. This paper tries to fill that gap with "stability experiments", which check how well a quantum error correction system can determine the product of a large region of stabilizers. Stability experiments achieve this by testing on a region that is locally a normal code but globally has a known product of stabilizers.
1 Introduction
Moving a logical observable through space requires combining stabilizer measurements into a byproduct operator, but there is no established benchmark for testing whether large products are computed reliably. Stability experiments test this ability using a spatial invariant with a known value, and can probe operational error-correction performance at small patch sizes.
1 Introduction
- A logical observable’s byproduct operator records whether that observable is negated, and stabilizer measurements determine how it changes when the observable moves.The relationship between byproduct operators for equivalent observables is determined by measurements of the stabilizers separating them.Source: §1 · pp. 1–2
- Moving a logical observable can require multiplying hundreds or millions of stabilizers, and any odd number of wrong measurements can give it the wrong sign.Such a logical error can silently produce bad results in the overlying algorithm.Source: §1 · p. 2
- Stability experiments address the lack of an established benchmark for directly verifying whether large stabilizer products are computed reliably.The paper contrasts this gap with memory experiments, a well-established benchmark for preserving a qubit through time [Rya+21] [Zha+22] [Kri+22] [And+20].Source: §1 · p. 2
- A stability experiment checks a known product of stabilizers across space, rather than a known invariant across time.Like memory experiments, it is degenerate in a computation because the expected result is known and the experiment would ordinarily be optimized away.Source: §1 · p. 2
- Stability experiments can quantify how many measurement rounds are needed for a target confidence and whether temporal code distance should differ from patch diameter.The paper notes there is no strict reason for timelike and spacelike code distances to be equal.Source: §1 · p. 2
- At small patch sizes, stability experiments may be easier than memory experiments and can show suppression even when the physical noise rate is above the surface-code threshold.The paper suggests this could demonstrate suppression of a relevant logical-error mechanism near, but not yet below, the surface-code threshold.Source: §1 · p. 3
2 The Stability Experiment
A stability experiment uses a surface-code patch with uniform boundary type to create a known global stabilizer product, then checks whether error correction determines it correctly. Its design requires a mismatch between spatial-boundary type and the data initialization and measurement basis, and repeated stabilizer measurements expose errors in the product.
2 The Stability Experiment
- A patch with the same boundary type around its perimeter has a known product of stabilizers because each data qubit contributes twice and the paired terms cancel.For the X-type example, the product of all X stabilizers is +1.Source: §2 · p. 3
- A chain of measurement errors can evade local detection events yet flip the global stabilizer product, causing the decoder to predict no corrections incorrectly.Figure 3 illustrates repeated misreporting of the same X stabilizer when data are initialized and measured in the Z basis.Source: §2 · p. 3, Figure 3 · p. 5
- The data initialization and measurement bases must differ from the spatial-boundary type; matching a timelike boundary to the spatial boundary makes the illustrated logical error detectable.In that matching case, data measurements provide an inviolable source of truth for correcting stabilizer measurements.Source: §2 · p. 3
- Choose a uniform-boundary patch, initialize and measure data in a basis that disagrees with that boundary type, and repeatedly measure the stabilizers.The patch size sets spatial scope, while the number of measurement repetitions can be chosen independently as the experiment’s code distance.Source: §2 · pp. 3–4
- Figure 2 relates stability experiments to a spacetime-rotated memory experiment and to spacelike portions of observable movement and parity measurement.The blue surfaces indicate valid parity sheets; red and black surfaces mark Z- and X-type boundaries, respectively.Source: Figure 2 · p. 4
- The decoder uses detection events to infer whether the product of selected stabilizer measurements was flipped; an incorrect prediction is a logical error.The product is computed from a round’s measurements corresponding to stabilizers matching the spatial-boundary type.Source: §2 · pp. 3–4
3 Simulation Results
Simulations tested whether stability experiments are more sensitive to measurement noise than memory experiments, but the standout result was their more forgiving finite-size threshold. In symmetric noise, stability curves also cross at higher physical error rates, with higher code distances reached by adding rounds rather than qubits.
Simulation Results
- The initial hypothesis was that stability experiments would be more sensitive to measurement errors than memory experiments, because their logical errors resemble repeated measurement errors.The comparison did not show a difference in measurement-error sensitivity large enough to stand out in error suppression.Source: §3 · p. 5
- The simulations varied measurement and other noise separately and compared fitted logical-error-rate slopes across patch diameters for memory and rounds for stability.Memory used 3x3, 5x5, and 7x7 patches; stability used 5, 15, and 25 rounds, with correlated minimum weight perfect matching for syndrome decoding.Source: §3 · p. 5
- The slope heatmap measures how much logical error improves as code distance increases, and whether physical noise is below the experiment’s effective threshold.For memory, code distance is patch diameter; for stability, it is the number of rounds.Source: §3 · p. 5, Figure 4 · p. 6
- Stability experiments were not visibly more sensitive to measurement error; instead, they had much more forgiving finite-size thresholds than memory experiments.A 4x4 stability experiment showed exponential suppression at physical noise rates roughly twice those needed by a two-round memory experiment.Source: §3 · pp. 5–6
- For symmetric noise, stability curves cross further right than memory curves, indicating a more forgiving effective threshold at small sizes.The plotted stability distances were 5, 15, and 25 rounds, versus memory patch diameters of 3, 5, and 7.Source: §3 · p. 6, Figure 5 · p. 7
4 Conclusion
The paper proposes stability experiments to verify a known global invariant across space and reports simulations showing a forgiving threshold at small patch sizes. It also notes that the experiment grows protection in only one dimension, while extending the approach beyond the surface code and advocating it as a benchmark.
4 Conclusion
- Simulations indicate that stability experiments offer a forgiving threshold and access to high code distances at small patch sizes.The experiment can quantify how many stabilizer-measurement rounds are needed to reach a target logical error rate when moving a logical qubit.Source: §4 · p. 8
- Stability experiments increase only the dimension that raises code distance, unlike computations that need protection to grow across two spatial dimensions and time.Widening and lengthening a patch increases the stabilizers in the product and requires lower physical error rates for the same logical error rate.Source: §4 · p. 8
- The experiment generalizes beyond the surface code to other error-correcting codes, including the honeycomb and color codes.Figures 6 and 7 show stability experiments in the honeycomb and color codes, respectively.Source: §4 · p. 8
- The paper argues that stability experiments should become a standard benchmark for error-correction systems.The proposed benchmark measures how reliably and efficiently systems determine larger stabilizer products.Source: §4 · p. 8
A Noise Model
The simulations use a noise model with two parameters: measurement-reset noise strength and unitary-operation noise strength. Table 2 defines the noise channels used by Table 1.
A Noise Model
- The simulation noise model has two parameters, measurement-reset noise strength pm and unitary-operation noise strength pu.Table 1 pairs gates in its left column with noisy compositions in its right column; A ◦ B = B · A, so B is applied after A.Source: Table 1 · p. 11
- Table 2 gives definitions of the noise channels used in Table 1.Source: Table 2 · p. 11
B Example Stability Experiment Circuit
The appendix identifies a noiseless Stim circuit for an example stability experiment and points to its source in the downloadable circuit archive.
B Example Stability Experiment Circuit
- The example is a noiseless Stim circuit from the stability circuit archive in the Zenodo dataset.It is identified by the circuit label “b=Z,d=4,pd=0.001,pm=0.001,r=25,type= stability.stim”.Source: B Example Stability Experiment Circuit · p. 12