Trapdoored Clifford Operators and Applications

AI summary

Overview

Research area: Quantum computing and cryptography, specifically quantum pseudorandomness, trapdoored operators, Clifford circuits, and worst-case-to-average-case reductions.

Technical level: Advanced. The paper assumes familiarity with the Clifford group, stabilizer tableaus, quantum polynomial-time indistinguishability, and learning parity with noise (LPN) assumptions.

Scope: The paper constructs trapdoored Clifford operator distributions that are computationally indistinguishable from uniformly random Cliffords yet can be sampled and implemented much faster with a hidden trapdoor, and it develops applications to quantum protocols and worst-case-to-average-case reductions.

What This Paper Is About

Uniformly random (n)-qubit Clifford operators are widely used in quantum computing, but sampling and implementing them requires near-quadratic resources: the optimal circuit size is (\Theta(n^{2}/\log n)), and a single application typically needs a nearly quadratic number of elementary gates. The paper asks whether this cost can be avoided by replacing true uniform randomness with pseudorandomness—distributions that no efficient quantum algorithm can distinguish from uniform—while hiding an efficient implementation behind a trapdoor. The goal is to build such “trapdoored Clifford operators” from cryptographic assumptions and show that they suffice for many protocols that rely on random Cliffords.

Key Contributions

1. Trapdoored Clifford operators from ring-LPN. Under the (dual) ring learning parity with noise assumption, the authors construct an (n)-qubit Clifford operator distribution whose samples are computationally indistinguishable from uniformly random Cliffords. A description and trapdoor can be sampled in (n^{1+\varepsilon}) time, and the trapdoor implements the operator using (n^{1+\varepsilon}) elementary gates and (n^{1+\varepsilon}) clean ancillary qubits for arbitrarily small constant (\varepsilon>0). Under a subexponential hardness assumption, these bounds improve to (n\cdot\operatorname{poly}\log(n)). The construction also supports near-linear evaluation of the conjugate action on any Pauli label and can be optimized to polylogarithmic depth under all-to-all connectivity.

2. Trapdoored invertible matrices over finite fields. The authors construct a trapdoored distribution over (M\in\operatorname{GL}(n,\mathbb{F}_{q})) that is samplable in (n^{1+\varepsilon}) time and whose trapdoor supports matrix–vector multiplication by (M), (M^{\mathsf{T}}), (M^{-1}), and (M^{-\mathsf{T}}) in (n^{1+\varepsilon}) time and polylogarithmic depth, improving to (n\cdot\operatorname{poly}\log(n)) under subexponential hardness. This resolves an open question left by Vaikuntanathan and Zamir [SODA’26]. The paper also constructs trapdoored triangular and symmetric matrices and shows that determinants of the sampled matrices can be computed efficiently, answering another question from that work.

3. Faster protocols based on random Cliffords. The trapdoored Cliffords can replace uniform random Cliffords in stabilizer simulation, direct fidelity estimation of Clifford channels, few-(T)-gate simulation, classical shadows, random stabilizer codes, and Clifford authentication. For authentication, the authors prove negligible soundness error against all adversarial behavior in Section 4.1, not only against efficient adversaries.

4. Worst-case-to-average-case reductions for linear algebra and Clifford problems. The constructions yield reductions for iterated matrix multiplication that work even when the given algorithm succeeds on a tiny fraction of inputs, and for determinant computation with error rate (1/2+\varepsilon) for sufficiently large (q). The paper also proves a worst-case-to-average-case reduction for Clifford circuit synthesis and shows that batching Clifford circuits—synthesizing a circuit for (C^{\otimes n}) from the tableau of a random (2n)-qubit Clifford (C)—is at least as hard as worst-case matrix multiplication, even when synthesis succeeds on a small constant fraction of random Cliffords or produces approximate implementations by general quantum circuits.

Main Findings

• Trapdoored Cliffords are pseudorandom and fast with a trapdoor. Samples are computationally indistinguishable from uniform Cliffords against quantum polynomial-time distinguishers, while the trapdoor enables implementation in (n^{1+\varepsilon}) gates and (n^{1+\varepsilon}) clean ancillas.

• Subexponential hardness gives quasi-linear efficiency. Assuming the strong (dual) ring-LPN assumption, sampling and implementation improve to (n\cdot\operatorname{poly}\log(n)).

• Pauli conjugation is near-linear. Given the trapdoor, conjugating a Pauli operator by the Clifford takes near-linear time, enabling faster classical simulation of stabilizer circuits and related tasks.

• Polylogarithmic depth is achievable. Under all-to-all connectivity, the Clifford circuit implementation can be done in polylogarithmic depth.

• Trapdoored matrices support multiplication by the matrix and its inverse. The distribution over (\operatorname{GL}(n,\mathbb{F}_{q})) supports efficient multiplication by (M), (M^{\mathsf{T}}), (M^{-1}), and (M^{-\mathsf{T}}), resolving the open question from Vaikuntanathan and Zamir. Trapdoored triangular and symmetric matrices are also constructed, and determinants can be computed efficiently.

• Batched Clifford synthesis is at least as hard as matrix multiplication. If an algorithm, given the tableau of a random (2n)-qubit Clifford (C), constructs a Clifford circuit for (C^{\otimes n}) with constant success probability, it can be converted into an algorithm that multiplies any two (n\times n) matrices over (\mathbb{F}_{2}) with high probability. This extends to approximate implementations that succeed on an arbitrarily small constant fraction of random Cliffords.

• Improved determinant reduction. For any (\varepsilon>0), the worst-case-to-average-case reduction for determinant computation works with error rate (1/2+\varepsilon) for sufficiently large (q).

• No empirical benchmarks or datasets are reported. The paper is theoretical; it does not include hardware experiments, dataset sizes, or benchmark comparisons.

Methodology in Plain English

The construction starts from a standard cryptographic idea: a public matrix (H=[I_{n}\mid C]), where (C) is a uniformly random circulant matrix, together with a sparse error matrix (E). The product (M=HE) looks uniformly random to an efficient observer under the ring-LPN assumption, but anyone who knows the sparse (E) and the structured (H) can multiply by (M) quickly. This gives a trapdoored matrix: the matrix looks random, but its hidden structure makes multiplication fast.

The authors then build triangular matrices recursively. A large strictly upper-triangular matrix is split into four blocks, with two smaller triangular matrices on the diagonal and a pseudorandom trapdoored matrix in the off-diagonal block. Recursing until the blocks are small enough to sample uniformly yields a distribution that is indistinguishable from a uniform triangular matrix. The same recursive block structure lets the trapdoor multiply quickly.

To obtain Clifford operators, the authors embed these trapdoored matrices into the Bravyi–Maslov Clifford sampler, which uses the Bruhat decomposition. The Bruhat decomposition expresses a Clifford operator as a product of triangular, symmetric, and permutation-like pieces. By making those pieces trapdoored, the paper obtains a Clifford operator that looks uniform but has an efficient hidden implementation. The same decomposition is used to support efficient inversion and determinant computation for the matrix constructions.

For applications, the authors substitute trapdoored Cliffords into protocols that repeatedly sample and apply random Cliffords. For the hardness reductions, they use the trapdoor to convert an average-case solver into a worst-case solver: the trapdoor lets them “randomize” a worst-case input while preserving the ability to recover the answer.

Why This Matters

Impact on research. The paper relaxes the requirement of true uniform randomness to computational pseudorandomness, which is sufficient for protocols involving only efficient quantum computations. It connects quantum pseudorandomness with classical LPN assumptions, resolves open questions on trapdoored matrices, and establishes new worst-case-to-average-case reductions for matrix and Clifford problems. The hardness result for batched Clifford synthesis links quantum circuit synthesis to classical matrix multiplication, which is a distinctive complexity-theoretic connection.

Potential applications. The paper is theoretical, but the constructions are motivated by practical quantum-computing tasks:

• Classical simulation and verification of quantum circuits. Near-linear Pauli conjugation speeds up stabilizer simulation, direct fidelity estimation of Clifford channels, and simulation of circuits with few (T) gates.

• Classical shadows and tomography. Trapdoored Cliffords can be used where random Cliffords are applied to an unknown state and measurement outcomes are recorded, supporting prediction of many efficiently computable observables, including fidelities with stabilizer states chosen after measurement.

• Random stabilizer codes. Applying a random Clifford to a state followed by zero ancillas gives a random stabilizer code, relevant to quantum secret sharing and erasure-resilient communication.

• Quantum authentication. The trapdoored construction improves the efficiency of Clifford authentication while retaining negligible soundness error against all adversaries in the proved setting.

• Gate-set shadow estimation and quantum data hiding. The paper notes these as further applications of the same template.

Industry relevance. The results are relevant to quantum software and hardware teams working on randomized benchmarking, error correction, and fault-tolerant compilation, where random Cliffords are routine but costly. They are also relevant to quantum cryptography and to algorithm designers interested in the hardness of circuit synthesis. The paper does not report an implementation or deployment; the relevance is conceptual and protocol-level.

Future Directions

• Weaken or diversify the cryptographic assumptions. The construction relies on the (dual) ring-LPN assumption and a strong subexponential variant for quasi-linear efficiency. It is natural to ask whether similar trapdoored Cliffords can be built from standard LPN, lattice assumptions, or other post-quantum assumptions.

• Quantify the design properties of trapdoored Cliffords. Uniform random Cliffords form an exact 3-design, while the trapdoored distribution may not. The paper argues that it still looks like an approximate 3-design to efficient algorithms; a precise characterization of which (k)-design properties survive would clarify the range of valid applications.

• Improve parameters and depth under realistic connectivity. The main construction uses (n^{1+\varepsilon}) ancillas and, for polylogarithmic depth, assumes all-to-all connectivity. Reducing ancilla requirements, achieving polylogarithmic depth under limited connectivity, and removing the subexponential assumption for quasi-linear time are natural next steps.

• Extend the hardness reductions. The batching result connects Clifford synthesis to matrix multiplication. Extending worst-case-to-average-case reductions to more Clifford problems, to approximate synthesis with general quantum circuits, and to other linear-algebraic tasks would further map the complexity landscape.

Target Audience

This paper is aimed at advanced researchers and graduate students in quantum cryptography, quantum complexity theory, and quantum pseudorandomness. It will be most useful to readers already comfortable with the stabilizer formalism, Clifford group structure, quantum polynomial-time indistinguishability, and LPN-style assumptions. Protocol designers and theoretical computer scientists interested in worst-case-to-average-case reductions will also benefit, while practitioners seeking implemented benchmarks or experimental results will not find them here.

Authors’ abstract

Random Clifford operators have numerous applications in quantum computing, including randomized benchmarking, classical shadows, and quantum authentication. However, sampling and implementing uniformly random $n$-qubit Clifford incur near-quadratic complexity due to the size of Clifford group. We introduce a cryptographic way to overcome these barriers: trapdoored Clifford operator distributions whose samples are computationally indistinguishable from uniformly random Cliffords, yet implementing them can be much faster given the trapdoor. We construct a distribution of trapdoored Clifford operators whose elements can be sampled and implemented in near-linear time under a variant of the learning parity with noise assumption. Our constructions allow fast tableau action on Pauli labels for classical simulation, and also can be optimized to admit polylogarithmic-depth implementation. Along the way, we construct trapdoored matrices over finite fields that support efficient multiplication by both a matrix and its inverse, resolving an open question left by Vaikuntanathan and Zamir [SODA'26]. We use these constructions to obtain faster protocols based on random Cliffords. We also explore their applications to the worst-case to average-case reductions for matrix and Clifford problems including the iterated matrix multiplication and Clifford circuit synthesis. In particular, we show the hardness of batching Clifford circuits: synthesizing circuits that apply the same Clifford to multiple registers is at least as hard as worst-case matrix multiplication, even when synthesis succeeds on a small constant fraction of random Cliffords. This extends to approximate implementations by general quantum circuits.