Subsystem surface and compass code sensitivities to non-identical infidelity distributions on heavy-hex lattice

The heavy hexagon code is a subsystem stabilizer code Poulin (2005) defined by the gauge group

	$\displaystyle G=\langle$	$\displaystyle X_{i,j}X_{i,j+1},Z_{i,j}Z_{i,j+1}Z_{i+1,j}Z_{i+1,j+1},$		(1)
		$\displaystyle Z_{2m-1,1}Z_{2m,1},Z_{2m,d}Z_{2m+1,d}\rangle$

where $i,j=1,2,\dots,d-1$, $m=1,2,\dots,\frac{d-1}{2}$, and $i+j$ is odd for the second term, as shown in Fig. 2a. We choose odd distances $d$ throughout this paper. The corresponding stabilizer group, illustrated in Fig. 2b, is

	$\displaystyle S=\langle$	$\displaystyle X_{i,j}X_{i,j+1}X_{i+1,j}X_{i+1,j+1},X_{1,2m}X_{1,2m+1},$		(2)
		$\displaystyle X_{d,2m-1}X_{d,2m},\prod_{j}Z_{i,j}Z_{i+1,j}\rangle$

with $i+j$ even for the first term. The X stabilizers are the same as the surface code whereas the Z stabilizers are those of the Bacon-Shor code. This code has a threshold of $p_{th}\sim 0.0045$ on the heavy hexagonal lattice for errors detected by the surface code stabilizers, and although there is no corresponding threshold for the Bacon-Shor code stabilizers, low logical error rates can still be achieved Chamberland et al. (2020). Quantum error correction has been experimentally demonstrated on the $d=3$ HHC Sundaresan et al. (2023).

In this work, we apply the idea of schedule-induced gauge-fixing Higgott and Breuckmann (2021) to optimize the total logical error probability of the HHC. This is a general idea that can be applied to CSS subsystem codes wherein gauge operators with deterministic eigenvalues are treated as stabilizers during syndrome processing. For the HHC, the circuits for each gauge operator measurement round are identical to Sundaresan et al. (2023), but the schedule may now repeat the same set of gauge operator measurements multiple times (see Appendix C for details). The decoding algorithm is also identical to the matching decoder in Sundaresan et al. (2023), but detection events are defined with respect to the instantaneous stabilizer groups as described in Higgott and Breuckmann (2021).

The subsystem surface code (SSC) Bravyi et al. (2013) and the standard surface code are related to each other by a local, constant depth stabilizer circuit. However, the gauge operators of the SSC have weight-3 or less and therefore can be measured by circuits that are naturally fault-tolerant. Higgott and Breuckmann have shown that subsystem surface codes can have high thresholds of $0.85\%$ under circuit model depolarizing noise Higgott and Breuckmann (2021), which exceeds the $0.67\%$ threshold of the surface code.

We focus on a rotated form of the SSC Brown et al. (2019) that is defined by the gauge group

	$\displaystyle G=\langle$	$\displaystyle Z_{i,j}Z_{i,j+1}Z_{t(i,j)},Z_{i+1,j}Z_{i+1,j+1}Z_{t(i,j)},$		(3)
		$\displaystyle Z_{2m-1,1}Z_{2m,1},Z_{2m,d}Z_{2m+1,d},$		(4)
		$\displaystyle X_{i,j}X_{i+1,j}X_{t(i,j)},X_{i,j+1}X_{i+1,j+1}X_{t(i,j)},$		(5)
		$\displaystyle X_{1,2m-1}X_{1,2m},X_{d,2m}X_{d,2m+1}\rangle,$		(6)

where $i,j=1,2,\dots,d-1$, $m=1,2,\dots,\frac{d-1}{2}$, and $i+j$ is odd for the weight-3 operators, as shown in Fig. 2c. The index $t(i,j)$ refers to the qubit in the center of the face with corners $(i,j)$, $(i+1,j)$, $(i,j+1)$, $(i+1,j+1)$, where $i+j$ is odd. A distance $d$ code has $(d-1)$ weight-2 X (Z) gauge operators and $(d-1)^{2}$ weight-3 X (Z) gauge operators. We use odd distances $d$ throughout the paper. The corresponding stabilizer group is

	$\displaystyle S=\langle$	$\displaystyle Z_{t(i,j-1)}Z_{i,j}Z_{i+1,j}Z_{i,j+1}Z_{i+1,j+1}Z_{t(i,j+1)},$		(7)
		$\displaystyle X_{t(i-1,j)}X_{i,j}X_{i,j+1}X_{i+1,j}X_{i+1,j+1}X_{t(i+1,j)}\rangle,$		(8)

where $i,j=1,2,\dots,d$ with $i+j$ is even. The operators $X_{i,j}$, $Z_{i,j}$, $X_{t(i,j)}$, and $Z_{t(i,j)}$ are defined to be identity whenever $i,j\notin\{1,2,\dots,d\}$. Example stabilizers, which are hexagonal in the bulk, are drawn in Fig. 2d. Since each pair of X and Z stabilizers that overlaps on four qubits has an associated pair of independent, anticommuting gauge operators, an RSSC with odd minimum distance $d$ encodes one logical qubit and $(d-1)^{2}/2$ gauge qubits into $d^{2}+(d-1)^{2}/2$ data qubits.

The RSSC is a promising option for implementing fault-tolerant quantum computing on a heavy hexagonal lattice for several reasons. First, the gauge operators can be measured by simple circuits. Second, the RSSC has a threshold, so it is expected to asymptotically out-perform the compass code. Finally, techniques for fault-tolerant computing with the surface code carry over to the subsystem surface code and differ only in implementation details.

The RSSC naturally overlays the heavy hexagonal lattice as shown in Fig. 1b. We refer to extra qubits for making or mediating measurements as measurement qubits. Each X gauge operator uses a single measurement qubit on the interior, and the weight-2 Z gauge operators on the boundary use three measurement qubits, assuming translation symmetry. Counting these additional $d^{2}+2d-3$ ancillary qubits, we operate a distance-$d$ code using a subset of the lattice containing $5d^{2}/2+d-5/2$ qubits.

Due to the natural mapping to the heavy hexagon lattice, the X gauge operators can be measured without extra gates using the circuit in Fig. 3a. Bit-flip errors on the measurement qubit propagate to at most one bit-flip error on the data modulo the gauge group, so there are no space-like hook errors. However the circuits can produce space-time hook errors from a single fault event, just like the surface code.

The Z gauge operators are more difficult to measure but each can be measured using the circuit in Fig. 3b. By inspecting all single-fault events, we conclude that this circuit has the property that any single fault gives rise to at most one X and one Z error on the data modulo the gauge group. Since these are correctable errors for CSS codes, the syndrome measurement circuit is fault-tolerant in the following sense. If $w$ faults occur then the output state deviates by at most $w$ X errors and $w$ Z errors modulo the gauge group. These X and Z errors can be corrected independently as long as there is no mechanism to convert them into Y errors.

On the heavy hexagon lattice, we find that the RSSC has a threshold of nearly $0.3\%$ for circuit-level noise. Additional details and a comparison with the HHC can be found in Appendix C.

It is worth mentioning other codes that can be directly implemented on the heavy hex layout, which are not included in this study. In particular, Floquet codes Hastings and Haah (2021), Floquet color codes Kesselring et al. (2022), matching codes Wootton (2022) and repetition codes Liepelt et al. (2023). In all these cases, syndrome measurements are made via two-qubit parity operators, for which the qubits on the edges of the heavy hex lattice can be used as auxiliaries.

Repetition codes are very much an outlier in this group since they protect only a logical bit rather than a logical qubit, and have a distance equal to the number of data qubits. We therefore do not include them in our study, since the results are likely be unrepresentative of truly quantum codes.

For Floquet codes, Floquet color codes and matching codes, though a direct implementation of these is possible on the heavy hex lattice, they are more ideally suited to systems in which direct parity measurements are possible Paetznick et al. (2023). Compared to this, implementation with the heavy hex would result in a significant decrease in threshold and performance, and significantly less resilience when there is a bias towards different noise sources Hetényi and Wootton (2023). It is for this reason that we do not include such codes in this study.

In this section we describe results from a parameterized error distribution for which the error rate is reciprocally related to a normally distributed random variable, $\mathbf{p}\propto 1/|\mathbf{X}|$. A physical motivation for this distribution is randomly distributed coherence times, like energy relaxation with a coherence time $T_{1}$. We assign an error rate $\mathbf{p}=(1-e^{\tau/|\mathbf{T_{1}}|})\sim\tau/|\mathbf{T_{1}}|$, where $\mathbf{T_{1}}$ is a random variable pulled from a normal distribution with mean, $\langle T_{1}\rangle$, standard deviation, $\sigma$ and $\tau$ is a constant representative of an effective gate time of the qubit operation. We simulate output error dependence on the standard deviation of the normal distribution, $\sigma$, for the RSSC code.

The logical output error rate dependence on uniform $p_{in}$ is shown in Fig. 4 (a) (see appendix C for more details) and the dependence on varying $\sigma$ is shown in Fig. 4 (b). We show the dependence normalized to the average and therefore vary a parameter $\alpha=\sigma/\langle T_{1}\rangle$. The average error rate in Fig. 4 (b) is $\mu=\tau/\langle T_{1}\rangle=10^{-3}$, an average cx error rate foreseeable in the near future  Stehlik et al. (2021). We show the logical error for both X or Z initialization and measurement cases. The output error rate shows an onset of increasing average logical error rate when $\sigma$ approaches $\alpha\sim 0.2-0.5$.

We now discuss the numerics in the context of a simple model that illustrates that the onset of increasing $\langle p_{out}\rangle$ may be understood as the onset of the mean of the input error rate, $\langle p_{in}\rangle$ increasing (i.e., $\mu\neq\langle p_{in}\rangle$). We first observe that numerical simulations of logical error rates for normally distributed error rates show no dependence on the standard deviation of the distribution, see appendix D, in contrast with the $\sigma$ dependence of the reciprocal normal distribution. To understand this observation, we note that a phenomenological model of the repetition code with normally distributed error rates results in an $\langle p_{out}^{rep}\rangle=3\bar{\epsilon}^{2}-2\bar{\epsilon}^{3}$ irrespective of the standard deviation of the error rates, see appendix E. Here we define an average output error of the phenomenological model, $\langle p_{out}^{rep}\rangle$, with contributing data sites at indices $i$. Each data site has normally distributed error rates, $\epsilon_{i}$ and an average error rate of all the data, $\bar{\epsilon}$. The phenomenological model provides the insight that random distributed error rates of a faulty type of component will result in an averaged contribution to the output error rate consistent with the numeric result. We conjecture that for a wide range of uncorrelated error rates the average logical error is a function of the average physical qubit error rate, $\langle p_{out}\rangle=F(\langle p_{in}\rangle)$, in contrast to dependence on higher moments of the distribution. On the other hand, we also note that although $\langle p_{out}^{rep}\rangle$ is not sensitive to changes in $\sigma_{\epsilon}$, the ’device-to-device’ variance of $\langle p_{out}^{rep}\rangle$ is predicted to be dependent on $\sigma_{\epsilon}$, see appendix E.

We return to the observation that the logical error rate increases for $\alpha\sim 0.2-0.3$ when the error rate distribution of the faulty locations is proportional to $\tau/|\mathbf{T_{1}}|$. The mean of such a distribution depends on the harmonic mean of the $\mathbf{T_{1}}$ distribution. The average is done for a normally distributed $T_{1}$ distribution, $P(\mathbf{T_{1}}=T_{1})=\frac{1}{\sqrt{2\pi\sigma}}e^{\frac{-(T_{1}-\langle T_{1}\rangle)^{2}}{2\sigma^{2}}}$, see appendix B for more details. The $T_{1}$ values sampled above versus below $\langle T_{1}\rangle$ are unevenly weighted and this leads to a $\sigma$ dependence of $\langle p_{in}\rangle$ despite $\langle T_{1}\rangle$ being independent of $\sigma$ consequently leading to an increase in $\langle p_{out}\rangle$.

This behavior is of general interest for any error process that depends reciprocally on a random process such as electronics noise impacting dephasing (e.g., $T_{2}$), although relative weights in the tails could lead to a deviation from the simple picture that the output error rate can be predicted by the average input error rates as if they were uniform error rates. Effects of outliers will be discussed in the following section.

In this section we examine the effect of changing error rate at a limited number of sites while the rest of the error rates for faulty locations are held constant. We ask, for example, how sensitive the logical error rate is to deviations of error rate at a single faulty location from the average?

We first examine the effect of increasing the error rates, $p_{in,0}$, of h, id and all cx operations that include the data qubit located at site 0, Fig. 1 (a). All other error rates are set to $5\times 10^{-4}$. A S-curve behavior is observed as $p_{in,0}$ is increased. While $p_{in,0}$ is similar to $\langle p_{in}\rangle$, little effect is observed on $p_{out}$, consistent with $\langle p_{in}\rangle$ not being significantly changed, see discussion above. At higher $p_{in,0}$, $p_{out}$ begins to rise and then saturates. Rapid increase in output error rate begins at an input error rate of $\sim 10^{-2}$ approximately 40 times larger than $\langle p_{in}\rangle$ for $d=5$. A single data qubit location can be relatively faulty compared to the devices average (i.e., $\langle p_{in}\rangle\sim 5\times 10^{-4}$) before the output error rate begins to be substantially degraded. The maximum increase in error is bounded further by the commensurate loss in distance of the code.

To further illustrate the effect of ’knocking out’ data qubits, we simulate output error as a function of uniform input error for instances where 0 to 4 data qubits are set to $p_{in}=0.5$, Fig. 5. We note that the sites 0 to 4 are along the logical Z operator in the RSSC. The measure X output error jumps after the addition of a single ’bad’ qubit at site 0 and ouput error increases quasi-linearly with the device $\langle p_{in}\rangle$, Fig. 5 (b). The end points of the sensitivity analysis of output error to varying the $p_{in,i}$ of a single ’bad’ qubit at site $i$, Fig. 5 (a), are identifiable in Fig. 5 (b) as indicated. We infer that there are similar S-curves between the other points in Fig. 5 (b).

Increasing the number of ’bad’ data qubits along the Z logical operator leads to monotonic jumps in output error rate. Qualitatively this could be interpreted as the sequential reduction of the effective distance of the code through ’knocking out’ code qubits along a logical operator. The 0 site is along both the logical X and Z operators. ’Knocking out’ the 0 site with a ’bad’ qubit reduces the correcting power of the logical qubit for both X and Z cases. The output error rate of the Z-measurement does not increase very rapidly, however. This behavior qualitatively may be understood as the code maintaining a $\sim 4$ distance correction for X-like errors combined with a lack of sensitivity of the logical Z-measure to logical-Z-like errors. Further discussion of site specific sensitivity can be found in appendix F.

Minimum weight perfect matching (MWPM) is used for decoding in the simulations. Weights for each edge in the decoder graph are assigned according to available information about error rates. We now investigate the decoder performance for the case that error rates for each gate operation are available, aware decoding (i.e., Dijkstra algorithm applied to unique edge weights in the decoding graph), in contrast to assuming an average error rate for the gate operations, naive (i.e., edge weights using the average error rate for i, h, cx operations).

An important way that decoding can improve is through resolving ambiguous syndromes for error chains greater than $(d+1)/2$. Random error rate distributions can offer ’tie breaking’ information (see for example appendix G). Improved error rates using aware decoding for random distributions for a phenomenological analysis have already been reported Tiurev et al. (2023). The prominence of ambiguous cases that can be resolved using this information is likely code dependent. To further probe the question of relative decoder effectiveness of aware decoding, we investigate the response of a heavy hex code (HHC) to these two different decoding approaches. The HHC is a hybrid of a surface code and compass code with flag qubits Chamberland et al. (2020); Sundaresan et al. (2023). This combination provides a useful relative view of the effectiveness of aware decoding for these two different codes.

Simulations were carried out for aware and naive and were done for the the $|0\rangle$ (Z) and $|+\rangle$ (X), four different cases, for several distances, Fig. 6. The Z basis corresponds to stronger sensitivity to the surface code like checks and the X basis corresponds to compass code like checks with flag qubits. In both X and Z bases, the aware decoder reduces $p_{out}$ compared to the naive case and the benefit increases with both decreasing $p_{in}$ and increasing $d$, consistent with predictions from phenomenological modeling Tiurev et al. (2023). The reduction in $p_{out}$ is, however, relatively weak for x basis cases and is in general relatively small for these small distance codes. Qualitatively weak improvements from decoder improvements for the X basis were reported in experiments on the $d=3$ HHC even when applying maximum likelihood methods. One contribution noted in that work was that the Z check circuit is susceptible to introducing a high rate of Z errors on the data, particularly because of how deflagging is done, which is difficult to improve with better decoding Sundaresan et al. (2023).

During the fabrication process of QPUs, checkpoints can be established where infidelities of qubit operations can be estimated Zhang et al. (2022). Imperfect fabrication processes lead to sets of QPUs each with their own estimated infidelity distributions. Ranking of the QPUs within these sets is useful for selection of which QPUs to ultimately deploy. A predictive pre-selection is desirable, all other considerations assumed to be the same.In the context of these findings, a paradigm of ranking by average input error rate of a dominant faulty circuit operations (e.g., two qubit gate operations) may therefore be an effective starting strategy in the absence of other supporting indicators like logical qubit error simulation.

This strategy might be further refined in the presence of a model for the logical error rate dependence on average input error rate, for example, circuit simulations of a particular code and device layout. In such circumstances, more quantitative evaluation of the sensitivity to differences in average input error, QPU to QPU, come into consideration including a better understanding of shot to shot standard deviation in the logical error rate.

The impact of one or a limited number of ’bad’ locations is a common question due to many factors ranging from fabrication imperfections to instability in device performance (e.g., spectral diffusion of two level systems Carroll et al. ). Here we loosely define a ’bad’ location as a faulty location for which the infidelity of the gate operation appears to be an outlier relative to the average error rate.

We have observed, in section IV.2, that the sensitivity of the logical error can be relatively weak to an outlier location until the error rate is appreciably larger than the contribution of the average error distribution. The ’bad’ location contribution to the logical error rate is, furthermore, limited as the error rate saturates at roughly the equivalent of the reduction of the code distance by one.

For a simplified order of magnitude estimation example, we might consider identification of ’bad’ two qubit gate locations by assuming a case where the two qubit gate locations are the dominant error rate for the logical error. Then a two qubit location is identifiable as a ’bad’ location when its error rate is: $\epsilon_{bad}\sim N_{q}\bar{\epsilon}$, where $N_{q}$ is the number of qubits in the encoding. In general the average error distribution has many contributions from the different circuit operations. Each of the operation types have their own relative contributions to the logical error rate.

Error rates are time dependent in realistic devices. There are a number of sources of time dependence Witzel et al. ; Müller et al. ; de Graaf et al. ; Krantz et al. (2019). A natural question is what logical error rates to expect in the presence of time dynamic noise. In many cases the time dynamics are correlated, which is out of the scope of this analysis. This work, however, does provide insight about the limit of uncorrelated time fluctuations. In this context, the observation that the average logical error rate is not dependent on higher moments of the distribution provides an indication that the logical error rate will converge to an average based on the average of the input error rates when uncorrelated noise processes are also stationary, while the variance of the logical error rates, in contrast, will depend on higher moments. It is left to future work to establish how close this limit approximates experimental cases that can have time and space correlations.

A device’s qubits performance relative to the pseudothreshold of a code is a predominant concern for design, fab and operation of quantum error correction on a QPU. Pseudothresholds are often estimated using simulations with forms of uniform error rates for types of circuit operations (e.g., two qubit operations). How to assess an actual device’s non-uniform infidelity distibution relative to a code’s pseudothreshold, particularly time varying distributions, without measuring or simulating the specific case (when measurement is not readily available) is a practical problem. This work provides the observation that for uncorrelated noise and surface-like codes, $\langle p_{out}\rangle=F(\bar{\epsilon})$ over a relative wide error rate range. The concept of a pseudothreshold therefore is also applicable when framed as an outcome of an average error rate of faulty locations in the error correction circuit. This observation may also be of utility in the context of system design, for which variability in system components are their own source of qubit operation infidelity distributions. System designs therefore will need to assess impact of outliers and variances of their component performances relative to their targets. Analysis based on the average error rates would greatly simplify the challenge of multi-distribution problems, in contrast to analysis of the contributions of multiple distribution each with their own multi-moment parameterizations.

Interest has increased recently in noisy connections between devices to form extended modular quantum error corrected patches Ramette et al. ; Auger ; Nickerson et al. . The impact of a sparse number of high error rate locations (i.e., connection points) on output logical error rates is of central interest in order to provide guidance about design and fabrication tolerances. The logical $\langle p_{out}\rangle$ dependence on ’bad’ qubit sites placed along a logical operator, in this work, is of tangential relevance as it highlights that if the intersecting ’seam’ between modules is placed along a logical operator, it may unnecessarily exaggerate the deleterious impact of the ’seam’ compared to staggering the intersections in a less spatially correlated mode (i.e., sparse random).