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 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 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.
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.
We give a pair of algorithms that efficiently learn a quantum state prepared by Clifford gates and O ( log n ) non-Clifford gates. Specifically, for an n -qubit state | ψ ⟩ prepared with at most t non-Clifford gates, our algorithms use p o l y ( n , 2 t , 1 / ε ) time and copies of | ψ ⟩ to learn | ψ ⟩ to trace distance at most ε .The first algorithm for this task is more efficient, but requires entangled measurements across two copies of | ψ ⟩ . The second algorithm uses only single-copy measurements at the cost of polynomial factors in runtime and sample complexity. Our algorithms more generally learn any state with sufficiently large stabilizer dimension, where a quantum state has stabilizer dimension k if it is stabilized by an abelian group of 2 k Pauli operators. We also develop an efficient property testing algorithm for stabilizer dimension, which may be of independent interest.
We consider the problem of testing whether an unknown n-qubit quantum state is a stabilizer state, with only single-copy access. We give an algorithm solving this problem using O(n) copies, and conversely prove that Ω(√n) copies are required for any algorithm. The main observation behind our algorithm is that when repeatedly measuring in a randomly chosen stabilizer basis, stabilizer states are the most likely among the set of all pure states to exhibit linear dependencies in measurement outcomes. Our algorithm is designed to probe deviations from this extremal behavior. For the lower bound, we first reduce stabilizer testing to the task of distinguishing random stabilizer states from the maximally mixed state. We then argue that, without loss of generality, it is sufficient to consider measurement strategies that a) lie in the commutant of the tensor action of the Clifford group and b) satisfy a Positive Partial Transpose (PPT) condition. By leveraging these constraints, together with novel results on the partial transposes of the generators of the Clifford commutant, we derive the lower bound on the sample complexity.
Pseudorandom states (PRSs) are state ensembles that cannot be efficiently distinguished from Haar-random states. However, the definition of PRSs has been limited to pure states and lacks robustness against noise. In this work, we introduce pseudorandom density matrices (PRDMs), ensembles of n -qubit states that are computationally indistinguishable from the generalized Hilbert-Schmidt ensemble (GHSE), which is constructed from ( n + m ) -qubit Haar random states with m qubits traced out. For m = 0 , PRDMs are equivalent to PRSs, whereas for m = ω ( log n ) , PRDMs are computationally indistinguishable from the maximally mixed state. PRDMs with m = ω ( log n ) are robust to unital noise channels and separated in terms of security from PRS. PRDMs can disguise valuable quantum resources as trivial states. In particular, we construct pseudoresource state ensembles, which possess near-maximal entanglement, magic and coherence, but are computationally indistinguishable from resource-free states. PRDMs exhibit a pseudoresource gap of Θ ( n ) vs 0, surpassing previously found gaps. We also render EFI pairs, a fundamental cryptographic primitive, robust to strong mixed unitary noise. Our work has major implications on quantum resource theory. We show that entanglement, magic, and coherence cannot be efficiently tested, and that black-box resource distillation requires a superpolynomial number of copies. We also establish lower bounds on the purity needed for efficient testing and black-box distillation. Finally, we introduce memoryless PRSs, a noise-robust notion of PRS, which are indistinguishable to Haar random states for efficient algorithms without quantum memory, as well as noise-robust quantum money. Our work provides a comprehensive framework of pseudorandomness for mixed states, which yields powerful quantum cryptographic primitives and fundamental bounds on quantum resource theories.