Low-Overhead Transversal Fault Tolerance for Universal Quantum Computation

TL;DR

The paper identifies the overhead bottleneck of fault-tolerant quantum computation arising from repeated syndrome extraction rounds. It introduces transversal algorithmic fault tolerance (TAFT) using CSS QLDPC codes, transversal Clifford gates, and correlated decoding with frame variables to achieve constant SE rounds per operation while preserving the correct logical measurement distribution. The authors prove a threshold and exponential suppression of logical errors with code distance, validate the approach with circuit-level simulations (including state distillation factories), and specialize the results to the surface code. They argue that TAFT can substantially reduce space-time overhead, potentially by over an order of magnitude, and discuss practical hardware paths, notably reconfigurable neutral-atom platforms, for realization.

Abstract

Fast, reliable logical operations are essential for realizing useful quantum computers. By redundantly encoding logical qubits into many physical qubits and using syndrome measurements to detect and correct errors, one can achieve low logical error rates. However, for many practical quantum error correcting (QEC) codes such as the surface code, due to syndrome measurement errors, standard constructions require multiple extraction rounds -- on the order of the code distance $d$ -- for fault-tolerant computation, particularly considering fault-tolerant state preparation. Here, we show that logical operations can be performed fault-tolerantly with only a constant number of extraction rounds for a broad class of QEC codes, including the surface code with magic state inputs and feed-forward, to achieve ``transversal algorithmic fault tolerance". Through the combination of transversal operations and novel strategies for correlated decoding, despite only having access to partial syndrome information, we prove that the deviation from the ideal logical measurement distribution can be made exponentially small in the distance, even if the instantaneous quantum state cannot be made close to a logical codeword due to measurement errors. We supplement this proof with circuit-level simulations in a range of relevant settings, demonstrating the fault tolerance and competitive performance of our approach. Our work sheds new light on the theory of quantum fault tolerance and has the potential to reduce the space-time cost of practical fault-tolerant quantum computation by over an order of magnitude.

Figures (8)

• Figure 1: Transversal algorithmic fault tolerance. (a) Conventional FT analysis separately examines each gadget (red boxes) in the circuit and ensures they are individually FT gottesman2010introductionshor1996faultaharonov1999fault. This typically requires $\Theta(d)$ syndrome extraction (SE) rounds to achieve FT. (b) Transversal algorithmic FT directly uses all accessible syndrome information up to a logical measurement (blue box), and guarantees FT of the logical measurement result, even if the gadgets are not individually FT and if future syndrome information is not yet accessible (partial decoding). We realize this through transversal operations, and only require a single SE round per logical operation.
• Figure 1: Surface code and transversal operations. (a) Illustration of the surface code. White circles indicate data qubits. Orange (green) plaquettes are $Z$ ($X$) stabilizers. The logical $\overline{Z}$ ($\overline{X}$) operator runs vertically (horizontally), and we choose our convention for fixing $Z$ ($X$) stabilizers to be performing a chain of $X$ ($Z$) flips to the left (bottom) boundary, as illustrated by the red line. (b) Illustration of transversal $\overline{H}$ gate, consisting of transversal $H$ gates followed by a reflection along the diagonal. Note that this differs from the usual transversal $\overline{H}$ gate, which applies a rotation in the second step. For the non-rotated surface code, both choices map $X$ ($Z$) stabilizers to $Z$ ($X$) stabilizers and hence are valid, but our choice leads to a smaller transversal partition size for the full circuit. (c) Illustration of transversal $\overline{S}$ gate, consisting of $S$ and $S^\dagger$ gates along the diagonal, together with $CZ$ gates between mirrored qubits.
• Figure 2: Illustration of fault tolerance strategy. (a) Illustration of the measurement distribution remaining invariant under the application of Pauli initialization errors. (b) Logical quantum circuit with measurement and feed-forward. All logical operations are transversal and interleaved with a single SE round, instead of $d$ SE rounds. We must decode and commit mid-circuit to a measurement result for the top qubit, despite lacking complete syndrome information on the remaining qubits (partial decoding). (c) With the measurement result of the top qubit, a feed-forward operation is applied and the circuit continues execution. Upon reaching a new logical measurement, decoding is performed again on the whole circuit. The new decoding round may assign a different result to the top qubit, causing an inconsistency in feed-forward operations. (d) To restore consistency and reliably interpret the logical measurement results given past measurements, we apply a $\overline{Z}$ operator on the $|{\overline{0}}\rangle$ initial state, which acts trivially on $|{\overline{0}}\rangle$, but changes the interpreted logical measurement results to be consistent with before. This also leads to a re-interpretation of the new logical measurement result. Inset: Illustration of a single syndrome measurement error in the surface code, which causes a Pauli initialization error between distinct decoding rounds.
• Figure 2: Illustration of error recovery and frame repair procedures. We illustrate the procedure for the surface code, where a cross-sectional view with one spatial axis and one time axis is shown. We only illustrate $X$ errors and $Z$ stabilizer measurement errors, which are relevant to interpreting the $\overline{Z}$ measurement. $X$ errors can terminate on orange boundaries, but cannot terminate on cyan boundaries. The transversal $\overline{CNOT}$ copies $X$ errors from the top to the bottom, resulting in a branching point (black cross) and an error cluster spanning both code blocks. (a) Error chains and frame flips. Chains of $X$-type errors (orange lines) lead to syndromes (end points) or terminate on appropriate boundaries. A line segment in the vertical direction is a data qubit $X$ error, while a line segment in the horizontal direction is a measurement error. Note that the $X$-type error cannot terminate on the transversal $Z$ measurement boundary. The random stabilizer initialization leads to a frame configuration on the logical $|{\overline{+}}\rangle$ initialization, as illustrated by the blue line and the flipped $Z$ stabilizer (blue point). This is similar to the frame stabilizer operator $g_s$ illustrated in ED Fig. \ref{['fig:surface_code_methods']}(a). (b) We first infer an error recovery operator, which has the same boundary as the error chain. Together, the error and recovery operator form the fault configuration, which triggers no detectors. We illustrate a few examples (orange lines) that do not lead to a logical error: (1) the fault configuration forms a closed loop and is equivalent to applying a stabilizer; (2) the fault configuration terminates on an initialization boundary; (3) the fault configuration terminates on an out-going, unmeasured logical qubit, but the forward-propagated errors onto the measured logical qubit are equivalent to a stabilizer. A logical error can only happen when the fault configuration spans across two opposing spatial boundaries (red line), which requires an error of weight $\Theta(d)$. (c,d) The frame repair operation returns the logical qubit to the code space with all stabilizers +1, corresponding to cancelling any residual flipped stabilizers on the initialization boundary. Note that the error recovery process may also lead to a change that needs to be accounted for by frame repair. An example choice of frame repair is shown in (c), which applies an overall $X$ operator on the logical measurement result. Alternatively, a different choice of frame repair shown in (d), related to the previous one by a frame logical flip, results in identity operation on the logical measurement result.
• Figure 3: Numerical verification of fault tolerance. (a) Simulation of circuit with repeated $\overline{ZZ}$ measurement (inset), where we commit mid-circuit to each measurement result of the logical ancilla using only the syndrome information up to that point. The total logical error rate as a function of circuit-level physical error rate $p$, for varying code distance $d$, shows clear threshold behavior. (b) Probability of inconsistent assignments of logical measurement results with and without the second step of our decoding strategy, as a function of code distance and for different physical error rates, for the same circuit as (a). Only with both steps do we observe exponential suppression of the logical error rate. (c) Comparison of two different methods for logical state preparation between three rotated surface codes and subsequent teleportation, for fixed circuit noise $p=0.3\%$. We use transversal gates (left) and lattice surgery (right), in both cases with only a single SE round. (d) With transversal gates, the error rate decreases exponentially with the code distance. With a single round of lattice surgery, the error rate instead increases linearly with code distance, as a single stabilizer measurement error affects the logical $\overline{ZZ}$ measurement result.