Certifying almost all quantum states with few single-qubit measurements

TL;DR

The paper introduces shadow overlaps as a rigorously analyzable, single-qubit-measurement–based method to certify that an experimentally prepared $n$-qubit state $ ho$ closely matches a target $|psi angle$, even for states with high entanglement. By linking certification to the mixing time of a random walk on the hypercube, the authors prove that almost all target states admit polynomial-time certification using $O(n^2)$ measurements (up to logarithmic factors), with structured states often admitting even faster mixing. The approach provides explicit sample-complexity bounds, a clear protocol (and a level-$m$ generalization) and a meaningful observable $L$ whose spectrum matches a Markov-transition matrix, enabling efficient prediction of nonlocal properties from certified representations. The work demonstrates broad applications to neural-network quantum-state tomography, benchmarking, and circuit optimization, with numerical demonstrations up to 120 qubits, and shows advantages over cross-entropy benchmarking in certain regimes. These results offer a scalable, measurement-efficient path to verify and utilize complex quantum states in practical settings.

Abstract

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).

Figures (5)

• Figure 1: Estimating the shadow overlap. Data collection phase: For each copy of the lab state $\rho$, a random qubit $\bm{k}$ is selected. All qubits except $\bm{k}$ are measured in the $Z$ basis. Qubit $\bm{k}$ is measured in a random $X$, $Y$, or $Z$ basis to obtain its classical shadow. Query phase: By querying the amplitudes of the target state $|\psi\rangle$ twice, the ideal post-measurement state $|\psi_{\bm{k}, \bm{z}}\rangle$ of qubit $\bm{k}$ is found. Using the classical shadow of qubit $\bm{k}$ from the lab state, its overlap $\bm{\omega}$ with $|\psi_{\bm{k}, \bm{z}}\rangle$ is evaluated. Finally, the shadow overlap $\mathop{\bf E}[\bm{\omega}]$ is estimated by averaging $\bm{\omega}$ across all copies.
• Figure 2: Neural network quantum state tomography: training and certifying a neural quantum state with the shadow overlap.(a) A dual-input neural network is trained to learn a quantum phase state \ref{['eq:randomPhaseStateML']} with random phases $\phi(x)$ on $n=120$ qubits using single qubit measurements. A shadow-based loss function trains the model on $50,000$ measurement data acquired as outlined in Protocol \ref{['proto:certification']}. The model is then certified using fidelity and shadow overlap on a separate data set of size $10,000$. (b) The trained neural quantum state is used to estimate the subsystem purity of the random phase state, exhibiting a high degree of entanglement compared to a randomly initialized neural quantum state.
• Figure 3: Benchmarking with the shadow overlap. The performance of the (normalized) shadow overlap, fidelity, and cross entropy benchmark (XEB) are compared in benchmarking noisy quantum states on 4 and 20 qubits for (a) a Haar random state subject to white noise, (b) a structured state, specifically a random phase state (generated from random product states) subjected to white noise, (c) a Haar random state with coherent noise, and (d) a random phase state with coherent noise. The error bars indicate statistical measurement errors, with shadow overlap displaying notably lower variance than XEB.
• Figure 4: Optimizing quantum circuits for state preparation. Training a low-depth quantum circuit consisting of Hadamard, controlled-$Z$, and $T$ gates to prepare a target state $|\psi\rangle$ given as a matrix product state (MPS). (a) As we approach the target state by building an appropriate circuit, the shadow overlap increases steadily with the number of circuit steps; in contrast, the fidelity sticks close to zero for many steps before growing abruptly. (b) Because the optimization landscape of fidelity has a barren plateau, training with fidelity fails to find a high-fidelity state-preparation circuit. In contrast, training with shadow overlap successfully finds a high-fidelity circuit.
• Figure 5: Various types of cycles $C^*$ for a given $(t',t,s,s')$. From left to right: when $s$ and $t$ are both good, $s$ is good but $t$ is bad, and $s$ and $t$ are both bad. The cycle formed by the green path and $(s,t)$ on the left has the following type: (1) the relative levels $L(v_i)-L(s)$ are $(0,1,2,1)$ in order and $|C^*|=4$, (2) $v_1=s, v_4=t$.

Theorems & Definitions (61)

• Theorem 1: Certification of quantum states, informal
• Theorem 2: Certification of almost all quantum states, informal
• Proposition 3
• proof
• Theorem 4
• proof
• Theorem 5: Sample Complexity of level-$m$ of Protocol \ref{['proto:verificationGeneral']}
• proof
• Theorem 6: Efficient certification using level-$m$ of Protocol \ref{['proto:verificationGeneral']}
• proof