Observation of a Fault Tolerance Threshold with Concatenated Codes

TL;DR

This work demonstrates a fault-tolerant scheme for concatenated CSS codes based on a generalized Shor code arranged in a butterfly network, achieving high threshold behavior with zero ancilla overhead and a tensor-network Bayesian decoder. The authors quantify state-preparation thresholds for erasure and unheralded noise, validate the approach on trapped-ion hardware, and show its applicability to quantum memory and universal gate synthesis via magic-state distillation and gate-injection. They introduce both optimal and approximate decoders, including a stacked probability passing method, and analyze extensions such as dynamical rewiring and code switching between encoding orders. The results indicate a viable path to universal quantum computation with relatively small base codes, offering potential advantages over LDPC-like codes in certain resource and overhead regimes, and providing scalable thresholds accessible on current devices.

Abstract

We introduce a fault-tolerant protocol for code concatenation of a generalized Shor code using a butterfly network architecture with high noise thresholds and low ancilla overhead to allow implementation on current devices. We develop a probability passing decoder using tensor networks that applies Bayesian updates to the marginal error probabilities after each layer of checks, achieving a state preparation threshold of $e_c \approx 0.089$ for erasure errors, and $\approx 0.015$ for unheralded noise. We implement our state preparation protocol on ion-trap hardware with added noise to demonstrate the threshold behavior in a real quantum device. We further theoretically test the performance of our scheme as a quantum memory and for universal quantum computation through the preparation of low-noise magic states for state distillation and $T$-gate injection.

Figures (23)

• Figure 1: (a) Example depth-4 encoding circuit of a $\mathrm{GS_{o}}$ code. In each layer, the left qubit entering a gate comes from the previous layer, while the right qubit is in a fresh pure state. One logical qubit (purple leg at root) is encoded into 16 physical qubits (leaves of tree). (b) Multitree (butterfly network) state preparation circuit for $|{\overline{{0}}}\rangle$ (purple legs all $|{0}\rangle$) or $|{\overline{{+}}}\rangle$ (purple legs all $|{+}\rangle$). Thick solid lines trace the tree that encodes the state on the "system," corresponding to the branches of the singletree in (a), while dashed gray lines indicate qubits used as ancillas. (c) Legend for (a) and (b). Blue and red circles denote CNOT (left qubit as control) and NOTC (right qubit as control), respectively. Each square node of the butterfly network consists of: (1) a "check gate" (CNOT for $|{\overline{{0}}}\rangle$, NOTC for $|{\overline{{+}}}\rangle$), (2) a measurement of the ancilla qubit in the corresponding basis ($Z$ or $X$), used to read off the syndrome in that layer, (3) resetting the ancilla to a fresh stabilizer state, (4) an encoding gate (CNOT for blue square nodes, NOTC for red square nodes). Each of these steps can incur errors, as described in the main text.
• Figure 2: Depth-2 multitree circuit for $|{\overline{{0}}}\rangle$, with a round of perfect stabilizer measurements appended to evaluate its performance. Highlighted wires show how a fault (yellow starburst) propagates through the circuit. The first round of checks (level 1 syndrome) distinguishes this fault from a bit flip on the second system qubit, which would propagate to a logically inequivalent error with the same level 2 syndrome.
• Figure 3: Logical failure probability of state preparation in classical simulations. (a) Erasure errors at rate $e$, optimal decoder. Left panel (i) shows system sizes $T=2$ to $T=11$, plotted on a log scale. Gray dashed line marks the threshold $e_c \approx 0.089$. Right panel (ii) is a scaling collapse to the form \ref{['eq:scaling']} with $e_c = 0.089, \nu = 2.85$, plotted on a linear scale with depths $T=3$-11. (b) Unheralded bit/phase flips at rate $r$, decoded via stacked probability passing. Gray dashed line marks the threshold $r_c \approx 0.015$. In all panels, the plotted quantity is averaged over $|{\overline{{0}}}\rangle$ and $|{\overline{{+}}}\rangle$. In this paper, all logarithms are base 10, and uncertainties are determined as the standard error of the mean across independent samples or the one sigma confidence interval across Bernoulli trials. We use a linear scale for the horizontal axis as our focus is on the near-threshold behavior, not the behavior near $p=0$.
• Figure 4: Logarithm of the failure probability on Quantinuum System Model H2, native noise + added heralded bit/phase flips at rate $e$ (\ref{['eq:herald-X']}) (a) and heralded coherent errors (\ref{['eq:coherent']}) at rate $q$ (b). The inset to (a) shows the subthreshold scaling at added noise rate $e=0.02$. In all panels, the plotted quantity is averaged over $|{\overline{{0}}}\rangle$ and $|{\overline{{+}}}\rangle$.
• Figure 5: (a) Steane syndrome measurement. Leftmost wire is an unknown logical state on the system. Yellow starbursts show the noise locations modeled in this work. Arrows indicate the flow of Pauli errors. (b) Depth-3 preparation circuit for an ancilla encoded in $|{\overline{{0}}}\rangle$, for use in the gadget shown in (a). This preparation circuit includes $X$ checks after the first layer and $Z$ checks after the second layer. Legend is the same as in \ref{['fig:model']}c. (c) Logical failure probability after (i) $2^{\lfloor T/2 \rfloor}$ rounds and (ii) 1 round of Steane syndrome measurement, under a bulk erasure model with optimal decoding. Apparent thresholds, achieving the best collapse to the scaling form \ref{['eq:scaling']}, are indicated by dashed gray lines. Uncertainties are determined as the standard error of the sample mean.