Automated Compilation Including Dropouts: Tolerating Defective Components in Stabiliser Codes

TL;DR

ACID tackles fabrication defects in quantum error correction by enabling ancilla-free, middle-out syndrome extraction via an ILP/CP-SAT-optimized contraction-schedule framework that adapts CSS stabiliser codes to defective hardware. It demonstrates substantial depth overhead reductions and resilience of the surface code compared with LUCI, and expands to bivariate bicycle and colour codes, highlighting broader applicability and hardware co-design implications. The approach preserves code-distance in many dropout scenarios and provides a concrete path toward scalable, fault-tolerant quantum memories on imperfect devices. Overall, ACID offers a practical route to increased chip yields and lower costs for quantum error-correcting hardware without sacrificing logical performance.

Abstract

Utility-scale solid-state quantum devices will need to fabricate quantum devices at scale using imperfect processes. By introducing tolerance to fabrication defects into the design of the quantum devices, we can improve the yield of usable quantum chips and lower the cost of useful systems. Automated Compilation Including Dropouts (ACID) is a framework that works in the ancilla-free (or `middle-out') paradigm, to generate syndrome extraction circuits for general stabiliser codes in the presence of defective couplers or qubits. In the ancilla-free paradigm, we do not designate particular qubits as measurement ancillas, instead measuring stabilisers using any of the data qubits in their support. This approach leads to a great deal of flexibility in how syndrome extraction circuits can be implemented. ACID works by constructing and solving an optimisation problem within the ancilla-free paradigm to find a short syndrome extraction circuit. Applied to the surface code, ACID produces syndrome-extraction circuits of depth between $1\times$ (no overhead) and $1.5\times$ relative to the depth of defect-free circuits. The LUCI algorithm, the best prior art, yielded a $2 \times$ overhead, so ACID offers a significant time saving. The yield of surface code chips with a logical error rate at most $10\times$ the dropout-free baseline is up to $3\times$ higher using ACID than using LUCI. I demonstrate the broad applicability of ACID by compiling syndrome extraction circuits for bivariate bicycle codes and the colour code. For these circuits, we incur a circuit-depth overhead of between $1\times$ (no overhead) and $2.5\times$ relative to defect-free circuits. I believe this work is the first to simulate both of these families of codes in the presence of fabrication defects.

Figures (17)

• Figure 1: A comparison between ACID and the state-of-the-art LUCI algorithm debroyLUCISurfaceCode2024 when both are applied to the distance-11 surface code. Top left: the proportion of cases where ACID and LUCI are able to produce syndrome extraction circuits with a logical error rate less than $10^{-6}$ for a distance-11 surface code implemented on a device with square-grid connectivity, assuming a random subset of qubits and couplers are defective. The numbers of defective qubits and defective couplers are shown on the $x$-axis, and the shaded region is the standard binomial error. Bottom left: In the same situations, the average number of 'contraction layers' (Section \ref{['sec:ancilla-free']}) required to construct a round of syndrome extraction, plotted with the standard error in the mean. LUCI always requires 4 rounds, whereas ACID is observed to never require more than 3. Top right and bottom right: The same comparison for the distance-11 surface code implemented on a device with hex-grid connectivity. These plots are a summary of the full data provided in Appendix \ref{['sec:plots']}.
• Figure 2: The ancilla-free approach to implementing quantum error-correcting codes.a. A single round of one possible syndrome extraction cycle for the triangular colour code, that consists of two contraction layers. I've highlighted three $X$-type (red) and three $Z$-type (teal) stabilisers and track the corresponding instantaneous stabilisers through the syndrome extraction cycle. We start (top-left) the cycle in the mid-cycle state, which comes from the standard definition of the colour code, and contains no ancilla qubits. In the first round, the three $X$ stabilisers and one of the $Z$ stabilisers are contracted, and the remaining two $Z$ stabilisers are expanded. We perform three timesteps' worth of CNOT gates to take us to the second stage, known as the $L_1$ end-cycle state. The contracted stabilisers are now each localised onto a single qubit which we measure in the appropriate basis. We undo the CNOTs to return to the mid-cycle state. In the second layer, we contract a different subset of the stabilisers (producing the $L_2$ end-cycle state), such that each stabiliser is measured at least once over the two rounds. Note that the $X$ stabiliser in the bottom-right is measured in both layers, but is contracted onto a different qubit in each. b. Here I show how a weight-six stabiliser with a cyclic local connectivity is affected by the presence of dropouts. In the top case, we have no dropouts and we arbitrarily choose a contraction schedule onto one of the qubits, subject only to compatibility with the schedules of neighbouring stabilisers. In the middle case, the presence of a dropped-out coupler forces us to use a different contraction schedule (note there are still two contraction schedules to choose from). In the bottom case, the presence of both a defective coupler and a defective qubit causes the remaining qubits in the stabiliser to become disconnected. In this case, we define 'quasi-stabilisers' from the remaining connected components and define product stabilisers and gauge qubits from them, forming a subsystem code. We then measure these stabilisers separately, and classically multiply the outcomes of multiple quasi-stabilisers to form deterministic detectors.
• Figure 3: An overview of the process of compiling a memory circuit around defects.
• Figure 4: The commutation relations of a CSS subsystem code (symplectic basis). The left and right 'columns' are each binary matrices, each of width $2n$ (twice the number of physical qubits) and height $n$, so that, taken together, the $2n$ rows form a complete basis for $F_2^{2n}$. Each row of either column represents a Pauli operator. Within each row the first $n$ bits represent the places where a Pauli operator contains an $X$, and the last $n$ bits represent where it contains a $Z$. Then the diagram can be read as a series of left-right pairs of Pauli operators, so that each operator anticommutes with its neighbour and commutes with all others. Within this representation, there are blocks corresponding to the parity check matrices $H^X$ and $H^Z$, the gauge operators $G^X$ and $G^Z$, and logical operators $L^X$ and $L^Z$. To complete the basis, I've included the destabiliser matrices $D^X$ and $D^Z$, although for practical purposes we do not need to compute these. The bulk of the stabilisers are the existing untouched stabilisers $U^X$ and $U^Z$, with the remaining stabilisers and gauge operators formed as products of the 'touched' quasi-stabilisers. Viewed this way, it is clear that a subsystem code can be seen as a stabiliser code with an arbitrary partition of the logical qubits into both logical qubits and gauge qubits. The other definition of a subsystem code, via a gauge group, is illustrated with a grey box. (Note - this representation ignores phases, which we would need to include if we were to write a full stabiliser tableau---see nielsenQuantumComputationQuantum2010)
• Figure 5: Detector assignment in the ancilla-free paradigm. Depicted is a portion of two rounds of syndrome extraction, each of which has $L=2$ layers. A detector is defined between two rounds, being contracted in the first layer of both. The detector includes the final measurement, but also measurements of the stabilisers expanded support in the intervening second layer.