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 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.
Bell sampling is a simple yet powerful measurement primitive that has recently attracted a lot of attention, and has proven to be a valuable tool in studying stabiliser states. Unfortunately, however, it is known that Bell sampling fails when used on qu\emph{d}its of dimension d>2. In this paper, we explore and quantify the limitations of Bell sampling on qudits, and propose new quantum algorithms to circumvent the use of Bell sampling in solving two important problems: learning stabiliser states and providing pseudorandomness lower bounds on qudits. More specifically, as our first result, we characterise the output distribution corresponding to Bell sampling on copies of a stabiliser state and show that the output can be uniformly random, and hence reveal no information. As our second result, for d=p prime we devise a quantum algorithm to identify an unknown stabiliser state in (Cp)⊗n that uses O(n) copies of the input state and runs in time O(n4). As our third result, we provide a quantum algorithm that efficiently distinguishes a Haar-random state from a state with non-negligible stabiliser fidelity. As a corollary, any Clifford circuit on qudits of dimension d using O(logn/logd) auxiliary non-Clifford single-qudit gates cannot prepare computationally pseudorandom quantum states.
It is well-known that learning a pure n-qubit stabilizer state ∣ψ⟩ both requires, and can be accomplished with, access to a number of copies of ∣ψ⟩ linear in n. However, the precise constant coefficient of this scaling does not appear to have been determined. Here we prove that Lδ(n), the smallest number of copies from which a quantum procedure can identify any stabilizer state with failure probability at most 0<δ<1/8, satisfies n+⌈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 n-qubit Clifford unitary from 2n+⌈log2(1/δ)⌉+4 queries, the n-dependence of which we show to be optimal. Our proof technique, which involves Fourier analysis on the abelian group 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 that quantum states with "low stabilizer complexity" can be efficiently distinguished from Haar-random. Specifically, given an n-qubit pure state |ψ⟩, we give an efficient algorithm that distinguishes whether |ψ⟩ is (i) Haar-random or (ii) a state with stabilizer fidelity at least 1/k (i.e., has fidelity at least 1/k with some stabilizer state), promised that one of these is the case. With black-box access to |ψ⟩, our algorithm uses O(k^{12} log(1/δ)) copies of |ψ⟩ and O(n k^{12} log(1/δ)) time to succeed with probability at least 1-δ, and, with access to a state preparation unitary for |ψ⟩ (and its inverse), O(k³ log(1/δ)) queries and O(n k³ log(1/δ)) time suffice. As a corollary, we prove that ω(log(n)) T-gates are necessary for any Clifford+T circuit to prepare computationally pseudorandom quantum states, a first-of-its-kind lower bound.
We prove that every pure state of stabilizer rank at most k has stabilizer extent at most 2O(klog(k+1)), and establish the analogous bound for the squared Clifford coefficient norm of Clifford-rank-k operators. This implies stabilizer fidelity at least 2−O(klog(k+1)), resolving the quantitative conjecture of (Mehraban-Tamasbi, STOC, 2025), and proves an 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 T-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/logn) lower bound for exact representations of n-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((logn)2/loglogn), respectively, a logn improvement over prior work (Kalra-Sinha, Quantum, 2026). We also obtain tomography algorithms for states of stabilizer rank at most k, with poly(n)2O(klog(k+1)) time and copy complexity, a nearly square-root improvement in the exponent over the best-known algorithm.
Bell sampling is a simple yet powerful tool based on measuring two copies of a quantum state in the Bell basis, and has found applications in a plethora of problems related to stabiliser states and measures of magic. However, it was not known how to generalise the procedure from qubits to d-level systems -- qudits -- for all dimensions d>2 in a useful way. Indeed, a prior work of the authors (arXiv'24) showed that the natural extension of Bell sampling to arbitrary dimensions fails to provide meaningful information about the quantum states being measured. In this paper, we overcome the difficulties encountered in previous works and develop a useful generalisation of Bell sampling to qudits of all d≥2. At the heart of our primitive is a new unitary, based on Lagrange's four-square theorem, that maps four copies of any stabiliser state ∣S⟩ to four copies of its complex conjugate ∣S∗⟩ (up to some Pauli operator), which may be of independent interest. We then demonstrate the utility of our new Bell sampling technique by lifting several known results from qubits to qudits for any d≥2: 1. Learning stabiliser states in O(n3) time with O(n) samples; 2. Solving the Hidden Stabiliser Group Problem in O~(n3/ε) time with O~(n/ε) samples; 3. Testing whether ∣ψ⟩ has stabiliser size at least dt or is ε-far from all such states in O~(n3/ε) time with O~(n/ε) samples; 4. Clifford circuits with at most n/2 single-qudit non-Clifford gates cannot prepare pseudorandom states; 5. Testing whether ∣ψ⟩ has stabiliser fidelity at least 1−ε1 or at most 1−ε2 with O(d2ε2/(ε2−ε1)2) samples if ε1=O(ε2/d2).
Stabilizer states are central to quantum computing, underlying quantum error correction, benchmarking, and efficient classical simulation. Yet their learnability exhibits a striking gap: an n-qubit stabilizer state can be learned from Θ(n) copies using two-copy Bell measurements, whereas non-adaptive single-copy measurements require Ω(n2) copies. Here we show that adaptivity completely closes this gap. We give a polynomial-time adaptive algorithm that learns an arbitrary n-qubit stabilizer state from Θ(n) single-copy Clifford measurements, matching the optimal sample complexity of Bell sampling without any multi-copy measurements. The same ideas yield a sample-optimal single-copy tolerant tester and, with k qubits of quantum memory, the optimal testing tradeoff Θ(n−k+1/ε) at infidelity ε. Finally, we show that this adaptive mechanism extends beyond exact stabilizer states: states of stabilizer nullity at most r, including states prepared by Clifford circuits with a bounded number of T gates, can be learned using O(n2r) single-copy measurements.