Peaked quantum advantage using error correction

TL;DR

The paper targets verifiable quantum advantage by introducing Hidden Code Sampling (HCS), a scheme that uses quantum error correction to create a conditionally peaked output distribution whose samples can be efficiently verified even when exact sampling is classically hard. The approach hinges on CSS codes to encode logical information and generate syndromes, enabling two complementary verification tests (PeakVerification and SyndromeVerification) that together resist efficient spoofing. It provides worst-case hardness results (via post-selection and $ extsf{GapP}$-hardness) and argues for plausible average-case hardness through connections to weight enumerators and code-phase transitions. A key contribution is the demonstration of a verification–simulation gap: while full simulation costs scale exponentially with system size, verification can be substantially cheaper under below-threshold noise, suggesting a pathway to practical, verifiable quantum advantage with near-term devices.

Abstract

A key issue of current quantum advantage experiments is that their verification requires a full classical simulation of the ideal computation. This limits the regime in which the experiments can be verified to precisely the regime in which they are also simulatable. An important outstanding question is therefore to find quantum advantage schemes that are also classically verifiable. We make progress on this question by designing a new quantum advantage proposal--Hidden Code Sampling--whose output distribution is conditionally peaked. These peaks enable verification in far less time than it takes for full simulation. At the same time, we show that exactly sampling from the output distribution is classically hard unless the polynomial hierarchy collapses, and we propose a plausible conjecture regarding average-case hardness. Our scheme is based on ideas from quantum error correction. The required quantum computations are closely related to quantum fault-tolerant circuits and can potentially be implemented transversally. Our proposal may thus give rise to a next generation of quantum advantage experiments en route to full quantum fault tolerance.

Figures (4)

• Figure 1: A small example illustrating the basics of error-correction through a simple $3$-qubit bitflip code. It proceeds in three steps. (a) We prepare the logical $\ket{\overline{1}}$ state of the code (the encoding circuit comprises the first two CNOT gates in the circuit). Observe that for the bitflip code, the logical $\ket{\overline{0}}$ state is the state $\ket{000}$ and the logical $\ket{\overline{1}}$ state is the state $\ket{111}$. (b) Then, after a bit-flip error, if we measure syndrome $11$ by introducing ancillas, we know that the logical registers are "peaked"---all the probability mass is in the state $\ket{010}$. (c) This peakedness is exactly the property we use to "decode," i.e. apply a sequence of (Pauli) operations revert the state back to $\ket{000}$. In this case, we apply the Pauli $X$ operation on the second qubit.
• Figure 2: The setup from Bob's perspective. He prepares the codestate $\ket{\overline +}$, which is given by $V \ket{+}^{\otimes k} \ket{+}^{\otimes (n-k_x)} \ket{0}^{\otimes (n-k_z)}$, where $V$ is a network of $\mathrm{CNOT}$ gates. Then he applies a layer of coherent rotations and then he samples by measuring in the $X$ basis.
• Figure 3: (a) T-gadget circuit, (b) Hadamard gadget circuit, (c) CZ gate using CNOT gates.
• Figure 4: The vertical axis is the relative entropy $D(q_{\rm ideal} || q_{\rm ref})$ between the ideal distribution and the Pauli spoofer. It is equal to the RED when the samples are drawn from the ideal distribution. The ideal score for random (3,9)-Gallager LDPC codes with code rate $2/3$ is given by the solid lines. The ideal score for random binary linear codes with rate $2/3$ is given by the dashed lines. Below a threshold rotation angle, as system size increases, the ideal score is empirically sample-efficient to compute and it tightly concentrates around a fixed non-zero value, which depends on what code we use.

Theorems & Definitions (23)

• Conjecture 1: Soundness of verification
• Theorem 2: Worst-case hardness
• proof
• Conjecture 3: Average-case hardness
• Theorem 4: Peakedness of the ideal distribution
• proof
• Lemma 5: Low-rank approximations of product channels gottesman_surviving_2024
• Lemma 6
• proof
• Proposition 7