Certifying that an n-qubit state synthesized in the lab is close to the target state is a fundamental task in quantum information science. However, existing rigorous protocols either require deep quantum circuits or exponentially many single-qubit measurements. In this work, we prove that almost all n-qubit target states, including those with exponential circuit complexity, can be certified from only O(n^2) single-qubit measurements. This result is established by a new technique that relates certification to the mixing time of a random walk. Our protocol has applications for benchmarking quantum systems, for optimizing quantum circuits to generate a desired target state, and for learning and verifying neural networks, tensor networks, and various other representations of quantum states using only single-qubit measurements. We show that such verified representations can be used to efficiently predict highly non-local properties that would otherwise require an exponential number of measurements. We demonstrate these applications in numerical experiments with up to 120 qubits, and observe advantage over existing methods such as cross-entropy benchmarking (XEB).
A central task in quantum information science is state certification: testing whether an unknown state is
ε1-close to a fixed target state, or
ε2-far. Recent work has shown that surprisingly simple measurement protocols--comprising only single-qubit measurements--suffice to certify arbitrary
n-qubit states [Huang, Preskill, Soleimanifar '25; Gupta, He, O'Donnell '25]. However, these certification protocols are not robust: rather than allowing constant
ε1, they can only positively certify states within
ε1=O(1/n) trace distance of the target. In many experimental settings, the appropriate error tolerance is constant as the system size grows, so this lack of robustness renders existing tests inapplicable at scale, no matter how many times the test is repeated.
Here we present robust certification protocols based on few-qubit measurements that apply to all but a
O(2−n)-fraction of pure target states. Our first protocol achieves constant robustness, i.e.
ε1=Θ(1), using a single
O(logn)-qubit measurement along with single-qubit measurements in the
Z or
X basis on the other qubits. As a corollary of its robustness, this protocol also achieves constant (in
n) copy complexity, which is optimal. Our second protocol uses exclusively single-qubit measurements and is nearly robust:
ε1=Ω(1/logn). Our tests are based on a new uncertainty principle for conditional fidelities, which may be of independent interest.
A fundamental challenge in quantum information science is certifying that an n-qubit state
ρprepared in the lab closely matches a target state
∣ψ⟩. Previous approaches to this problem often require deep quantum circuits, exponentially many single-qubit measurements, or are limited to specific state families. In this work, we introduce a new method that leverages a connection between state certification and the mixing time of a random walk, allowing almost all n-qubit target states, including those with exponential circuit complexity, to be certified with only
O(n2)single-qubit measurements. Our protocol is broadly compatible with various experimental platforms and has applications in benchmarking quantum systems, optimizing quantum circuits, and efficiently learning and verifying representations of quantum states—such as neural networks and tensor networks—using only single-qubit measurements. Moreover, these verified representations enable the efficient prediction of highly non-local properties of
ρthat would otherwise require an exponential number of measurements.
The generation of certifiable randomness is the most fundamental information-theoretic task that meaningfully separates quantum devices from their classical counterparts. We propose a protocol for exponential certified randomness expansion using a single quantum device. The protocol calls for the device to implement a simple quantum circuit of constant depth on a 2D lattice of qubits. The output of the circuit can be verified classically in linear time, and is guaranteed to contain a polynomial number of certified random bits assuming that the device used to generate the output operated using a (classical or quantum) circuit of sub-logarithmic depth. This assumption contrasts with the locality assumption used for randomness certification based on Bell inequality violation and more recent proposals for randomness certification based on computational assumptions. Furthermore, to demonstrate randomness generation it is sufficient for a device to sample from the ideal output distribution within constant statistical distance. Our procedure is inspired by recent work of Bravyi et al. (Science 362(6412):308-311, 2018), who introduced a relational problem that can be solved by a constant-depth quantum circuit, but provably cannot be solved by any classical circuit of sub-logarithmic depth. We develop the discovery of Bravyi et al. into a framework for robust randomness expansion. Our results lead to a new proposal for a demonstrated quantum advantage that has some advantages compared to existing proposals. First, our proposal does not rest on any complexity-theoretic conjectures, but relies on the physical assumption that the adversarial device being tested implements a circuit of sub-logarithmic depth. Second, success on our task can be easily verified in classical linear time. Finally, our task is more noise-tolerant than most other existing proposals that can only tolerate multiplicative error, or require additional conjectures from complexity theory; in contrast, we are able to allow a small constant additive error in total variation distance between the sampled and ideal distributions.
A central task in quantum information science is state certification: testing whether an unknown state is є1-close to a fixed target state, or є2-far. Recent work has shown that surprisingly simple measurement protocols – comprising only single-qubit measurements – suffice to certify arbitrary n-qubit states. However, these certification protocols are not robust: rather than allowing constant є1, they can only positively certify states within є1=O(1/n) trace distance of the target. In many experimental settings, the appropriate error tolerance is constant as the system size grows, so this lack of robustness renders existing tests inapplicable at scale, no matter how many times the test is repeated.
Characterizing increasingly complex quantum systems is a central task in quantum information science, yet experimental costs often scale prohibitively with system size. Certifying key properties using simple local measurements is highly desirable but challenging. In this work, we introduce a highly general certification framework based on a physical phenomenon that we call localizable quantumness: for generic many-body states, essential quantum properties are robustly preserved within the projected ensembles on small subsystems after performing local projective measurements on the rest of the system. Leveraging this insight, we develop protocols to certify global properties—including multipartite entanglement, circuit complexity, and quantum magic—by witnessing them on a small, accessible subsystem. Remarkably, randomizing the local measurement bases extends this capability to certify state fidelity. Relying solely on local Pauli measurements, these protocols achieve constant sample complexity and robustness for almost all quantum states, including a wide range of physically relevant states. For certifying the fidelity of n -qubit states, this O ( 1 ) scaling dramatically improves upon state-of-the-art protocols requiring O ( n 4 ) samples. Our unified framework provides both a practical toolkit for large-scale quantum certification and a novel lens into complex many-body systems.
A fundamental task in quantum information science is state certification: testing whether a lab-prepared n-qubit state is close to a given hypothesis state. In this work, we show that every pure hypothesis state can be certified using only O(n^2) single-qubit measurements applied to O(n) copies of the lab state. Prior to our work, it was not known whether even subexponentially many single-qubit measurements could suffice to certify arbitrary states. This resolves the main open question of Huang, Preskill, and Soleimanifar (FOCS 2024, QIP 2024).