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arXiv:2410.24202 · ACM Symposium on Theory of Computing (STOC)

Improved Bounds for Testing Low Stabilizer Complexity States

Saeed Adel Mehraban, Mehrdad Tahmasbi

Published Jun 15, 2025• 1 citation
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Abstract

Stabilizer states are fundamental families of quantum states with crucial applications such as error correction, quantum computation, and simulation of quantum circuits. In this paper, we study the problem of testing how close or far a quantum state is to a stabilizer state. We make two contributions: First, we improve the state-of-the-art parameters for the tolerant testing of stabilizer states. In particular, we show that there is an efficient quantum primitive to distinguish if the maximum fidelity of a quantum state with a stabilizer state is $\geq ε_1$ or $\leq ε_2$, given one of them is the case, provided that $ε_2 \leq ε_1^{O(1)}$. This result improves the parameters in the previous work [AD24] which assumed $ε_2 \leq e^{- 1/ε^{O(1)}_1}$ [AD24]. Our proof technique extends the toolsets developed in [AD24] by applying a random Clifford map which balances the characteristic function of a quantum state, enabling the use of standard proof techniques from higher-order Fourier analysis for Boolean functions [HHL19, Sam07], where improved testing bounds are available. Second, we study the problem of testing low stabilizer rank states. We show that if for an infinite family of quantum states stabilizer rank is lower than a constant independent of system size, then stabilizer fidelity is lower bounded by an absolute constant. Using a result of [GIKL22], one of the implications of this result is that low approximate stabilizer rank states are not pseudo-random. At the same time our work was completed and posted on arXiv, two other groups [BvDH24, ABD24] independently achieved similar exponential to polynomial improvements for tolerant testing, each using a different approach.

DOI
10.1145/3717823.3718228

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Stabilizer states are fundamental families of quantum states with crucial applications such as error correction, quantum computation, and simulation of quantum circuits. In this paper, we study the problem of testing how close or far a quantum state is to a stabilizer state. We make two contributions: First, we improve the state-of-the-art parameters for the tolerant testing of stabilizer states. In particular, we show that there is an efficient quantum primitive to distinguish if the maximum fidelity of a quantum state with a stabilizer state is ≥ε1\geq ε_1≥ε1​ or ≤ε2\leq ε_2≤ε2​, given one of them is the case, provided that ε2≤ε1O(1)ε_2 \leq ε_1^{O(1)}ε2​≤ε1O(1)​. This result improves the parameters in the previous work [AD24] which assumed ε2≤e−1/ε1O(1)ε_2 \leq e^{- 1/ε^{O(1)}_1}ε2​≤e−1/ε1O(1)​ [AD24]. Our proof technique extends the toolsets developed in [AD24] by applying a random Clifford map which balances the characteristic function of a quantum state, enabling the use of standard proof techniques from higher-order Fourier analysis for Boolean functions [HHL19, Sam07], where improved testing bounds are available. Second, we study the problem of testing low stabilizer rank states. We show that if for an infinite family of quantum states stabilizer rank is lower than a constant independent of system size, then stabilizer fidelity is lower bounded by an absolute constant. Using a result of [GIKL22], one of the implications of this result is that low approximate stabilizer rank states are not pseudo-random. At the same time our work was completed and posted on arXiv, two other groups [BvDH24, ABD24] independently achieved similar exponential to polynomial improvements for tolerant testing, each using a different approach.
We study the complexity of learning quantum states in various models with respect to the stabilizer formalism and obtain the following results: We prove that Ω(n) T-gates are necessary for any Clifford+T circuit to prepare computationally pseudorandom quantum states, an exponential improvement over the previously known bound. This bound is asymptotically tight if linear-time quantum-secure pseudorandom functions exist. Given an n-qubit pure quantum state |ψ⟩ that has fidelity at least τ with some stabilizer state, we give an algorithm that outputs a succinct description of a stabilizer state that witnesses fidelity at least τ − ε. The algorithm uses O(n/(ε2τ4)) samples and exp(O(n/τ4)) / ε2 time. In the regime of τ constant, this algorithm estimates stabilizer fidelity substantially faster than the naive exp(O(n2))-time brute-force algorithm over all stabilizer states. In the special case of τ > cos2(π/8), we show that a modification of the above algorithm runs in polynomial time. We exhibit a tolerant property testing algorithm for stabilizer states. The underlying algorithmic primitive in all of our results is Bell difference sampling. To prove our results, we establish and/or strengthen connections between Bell difference sampling, symplectic Fourier analysis, and graph theory.
We prove that every pure state of stabilizer rank at most kkk has stabilizer extent at most 2O(klog⁡(k+1))2^{O(\sqrt{k\log(k+1)})}2O(klog(k+1)​), and establish the analogous bound for the squared Clifford coefficient norm of Clifford-rank-kkk operators. This implies stabilizer fidelity at least 2−O(klog⁡(k+1))2^{-O(\sqrt{k\log(k+1)})}2−O(klog(k+1)​), resolving the quantitative conjecture of (Mehraban-Tamasbi, STOC, 2025), and proves an Ω(n2/log⁡n)Ω(n^2/\log n)Ω(n2/logn) lower bound for the approximate stabilizer rank of tensor powers of any non-stabilizer qubit state. The latter result generalizes the best-known lower bound for tensor powers of TTT-states (Mehraban-Tamasbi, STOC, 2024) to arbitrary non-stabilizer qubit states, including magic states. As a consequence of the Clifford rank--norm inequality, we obtain an Ω(n2/log⁡n)Ω(n^2/\log n)Ω(n2/logn) lower bound for exact representations of nnn-bit AND function by quadratic phases, improving the previous best-known linear bound. Further consequences rule out pseudorandom state and unitary ensembles with approximate stabilizer and Clifford rank O((log⁡n)2/log⁡log⁡n)O((\log n)^2/\log\log n)O((logn)2/loglogn), respectively, a log⁡n\log nlogn improvement over prior work (Kalra-Sinha, Quantum, 2026). We also obtain tomography algorithms for states of stabilizer rank at most kkk, with poly(n)2O(klog⁡(k+1))poly(n)2^{O(\sqrt{k\log(k+1)})}poly(n)2O(klog(k+1)​) time and copy complexity, a nearly square-root improvement in the exponent over the best-known algorithm.
The stabilizer rank of a quantum state ψ is the minimal r such that |ψ⟩=∑j=1rcj|φj⟩ for cj∈C and stabilizer states φj . The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n -th tensor power of single-qubit magic states.We prove a lower bound of Ω(n) on the stabilizer rank of such states, improving a previous lower bound of Ω(n) of Bravyi, Smith and Smolin \cite{BSS16}. Further, we prove that for a sufficiently small constant δ , the stabilizer rank of any state which is δ -close to those states is Ω(n/log⁡n) . This is the first non-trivial lower bound for approximate stabilizer rank.Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of F2n , and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.
The approximate stabilizer rank of a quantum state is the minimum number of terms in any approximate decomposition of that state into stabilizer states. Bravyi and Gosset showed that the approximate stabilizer rank of a so-called “magic” state like |T⟩⊗ n, up to polynomial factors, is an upper bound on the number of classical operations required to simulate an arbitrary quantum circuit with Clifford gates and n number of T gates. As a result, an exponential lower bound on this quantity seems inevitable. Despite this intuition, several attempts using various techniques could not lead to a better than a linear lower bound on the “exact” rank of |T⟩⊗ n, meaning the minimal size of a decomposition that exactly produces the state. For the “approximate” rank, which is more realistically related to the cost of simulating quantum circuits, no lower bound better than Ω(√n) has been known. In this paper, we improve the lower bound on the approximate rank to Ω(n2) for a wide range of the approximation parameters. An immediate corollary of our result is the existence of polynomial time computable functions which require a super-linear number of terms in any decomposition into exponentials of quadratic forms over F2, resolving a question by Williams. Our approach is based on a strong lower bound on the approximate rank of a quantum state sampled from the Haar measure, a step-by-step analysis of the approximate rank of a magic-state teleportation protocol to sample from the Haar measure, and a result about trading Clifford operations with T gates.
It is well-known that learning a pure nnn-qubit stabilizer state ∣ψ⟩|ψ\rangle∣ψ⟩ both requires, and can be accomplished with, access to a number of copies of ∣ψ⟩|ψ\rangle∣ψ⟩ linear in nnn. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that Lδ(n)L_δ(n)Lδ​(n), the smallest number of copies from which a quantum procedure can identify any stabilizer state with failure probability at most 0<δ<1/80<δ<1/80<δ<1/8, satisfies n+⌈log⁡2(1/δ)⌉−3≤Lδ(n)≤n+⌈log⁡2(1/δ)⌉+4n+\lceil\log_2(1/δ)\rceil-3\leq L_δ(n)\leq n+\left\lceil\log_2(1/δ)\right\rceil+4n+⌈log2​(1/δ)⌉−3≤Lδ​(n)≤n+⌈log2​(1/δ)⌉+4. We present a polynomial-time quantum learning algorithm that saturates this bound, achieving a constant factor improvement in sample-complexity over previously known approaches. As an immediate corollary, we obtain via the Choi-Jamiolkowski isomorphism an algorithm for learning an unknown nnn-qubit Clifford unitary from 2n+⌈log⁡2(1/δ)⌉+42n+\left\lceil\log_2(1/δ)\right\rceil+42n+⌈log2​(1/δ)⌉+4 queries, the nnn-dependence of which we show to be optimal. Our proof technique, which involves Fourier analysis on the abelian group Z4n×F2n(n−1)/2\mathbb{Z}_4^n \times \mathbb{F}_2^{n(n-1)/2}Z4n​×F2n(n−1)/2​, seems to be qualitatively different to previous approaches to stabilizer state learning, and may be of some independent interest; in particular, it admits natural generalisations to further problems in quantum learning theory.
We show an improved inverse theorem for the Gowers-333 norm of nnn-qubit quantum states ∣ψ⟩|ψ\rangle∣ψ⟩ which states that: for every γ≥0γ\geq 0γ≥0, if the Gowers(∣ψ⟩,3)8≥γ\textsf{Gowers}(|ψ\rangle,3)^8 \geq γGowers(∣ψ⟩,3)8≥γ then the stabilizer fidelity of ∣ψ⟩|ψ\rangle∣ψ⟩ is at least γCγ^CγC for some constant C>1C>1C>1. This implies a constant-sample polynomial-time tolerant testing algorithm for stabilizer states which accepts if an unknown state is ε1\varepsilon_1ε1​-close to a stabilizer state in fidelity and rejects when ∣ψ⟩|ψ\rangle∣ψ⟩ is ε2≤ε1C\varepsilon_2 \leq \varepsilon_1^Cε2​≤ε1C​-far from all stabilizer states, promised one of them is the case.