We study efficient quantum certification algorithms for quantum state set and unitary quantum channel. We present an algorithm that uses $O(\varepsilon^{-4}\ln |\mathcal{P}|)$ copies of an unknown state to distinguish whether the unknown state is contained in or $\varepsilon$-far from a finite set $\mathcal{P}$ of known states with respect to the trace distance. This algorithm is more sample-efficient in some settings. Previous study showed that one can distinguish whether an unknown unitary $U$ is equal to or $\varepsilon$-far from a known or unknown unitary $V$ in fixed dimension with $O(\varepsilon^{-2})$ uses of the unitary, in which the Choi state is used and thus an ancilla system is needed. We give an algorithm that distinguishes the two cases with $O(\varepsilon^{-1})$ uses of the unitary, using much fewer or no ancilla compared with previous results.
We study efficient quantum certification algorithms for quantum state set and unitary quantum channel. We present an algorithm that uses O(ε−4ln∣P∣) copies of an unknown state to distinguish whether the unknown state is contained in or ε-far from a finite set P of known states with respect to the trace distance. This algorithm is more sample-efficient in some settings. Previous study showed that one can distinguish whether an unknown unitary U is equal to or ε-far from a known or unknown unitary V in fixed dimension with O(ε−2) uses of the unitary, in which the Choi state is used and thus an ancilla system is needed. We give an algorithm that distinguishes the two cases with O(ε−1) uses of the unitary, using much fewer or no ancilla compared with previous results.
We study efficient quantum certification algorithms for unitary quantum process using no ancilla. Previous study showed that one can distinguish whether an unknown unitary U is equal to or ε-far from a known or unknown unitary V in fixed dimension with O(ε−2) uses of the unitary, in which the Choi state is used and thus a high dimensional ancilla system is always needed. We give an algorithm that distinguishes the two cases with O(ε−1) uses of the unitary, using fewer or no ancilla, outperforming previous relevant results.
In the problem of quantum channel certification, we have black box access to a quantum process and would like to decide if this process matches some predefined specification or is ε-far from this specification. The objective is to achieve this task while minimizing the number of times the black box is used. Here, we focus on optimal incoherent strategies for two relevant extreme cases of channel certification. The first one is when the predefined specification is a unitary channel, e.g., a gate in a quantum circuit. In this case, we show that testing whether the black box is described by a fixed unitary operator in dimension d or ε-far from it in the trace norm requires Θ(d/ε2) uses of the black box. The second setting we consider is when the predefined specification is a completely depolarizing channel with input dimension din and output dimension dout. In this case, we prove that, in the non-adaptive setting, Θ~(din2dout1.5/ε2) uses of the channel are necessary and sufficient to verify whether it is equal to the depolarizing channel or ε-far from it in the diamond norm. Finally, we prove a lower bound of Ω(din2dout/ε2) for this problem in the adaptive setting. Note that the special case din=1 corresponds to the well-studied quantum state certification problem.
We study the problems of quantum state certification, equivalence testing and independence testing. In certification, given samples of an unknown quantum state ρ and the description of a state σ, the goal is to test whether ρ=σ, or whether ρ and σ are far in a given distance measure. In equivalence testing, σ is also unknown and only accessible via samples. Independence testing decides whether ρAC=ρA⊗ρC, or is far from being a product. The sample complexities of these problems are now well-understood for a decision gap ε in trace distance: in the single-copy measurement setting with d-dimensional states, all three tasks can be solved using the same non-adaptive approach, which uses Θ(d3/2/ε2) samples and is optimal in general, even without adaptivity.
In this work, we consider decision gaps expressed in fidelity and study possible separations between these problems and how adaptivity can help. We prove that certification with respect to fidelity for a state σ of rank r does not benefit from adaptivity and requires Θ(r3/2/ε) samples. For equivalence testing and independence testing, we provide adaptive algorithms using O(min{d3/2/ε2,d9/4/ε}) and O(min{(dAdC)3/2/ε2,dA9/4dC3/4/ε}) samples, for dA≥dC, respectively. Our main technique is a framework that uses partial learning and a reduction to testing in ℓ2-distance, adapted from the distribution testing literature. We show that adaptivity matters for equivalence testing in fidelity by proving that Ω(1/ε2) samples are necessary in the non-adaptive case even for qubits, showing a separation from certification.
We consider the problem of quantum state certification, where one is given n copies of an unknown d-dimensional quantum mixed state ρ, and one wants to test whether ρ is equal to some known mixed state σ or else is є-far from σ. The goal is to use notably fewer copies than the Ω(d2) needed for full tomography on ρ (i.e., density estimation). We give two robust state certification algorithms: one with respect to fidelity using n = O(d/є) copies, and one with respect to trace distance using n = O(d/є2) copies. The latter algorithm also applies when σ is unknown as well. These copy complexities are optimal up to constant factors.
We consider the problem of quantum channel certification to unitary, where one is given access to an unknown d-dimensional channel E, and wants to test whether E is equal to a target unitary channel or is ε-far from it in the diamond norm. We present optimal quantum algorithms for this problem, settling the query complexities in three access models with increasing power. Specifically, we show that:
(i) Θ(d/ε2) queries suffice for incoherent access model, matching the lower bound due to Fawzi, Flammarion, Garivier, and Oufkir (COLT 2023).
(ii) Θ(d/ε) queries suffice for coherent access model, matching the lower bound due to Regev and Schiff (ICALP 2008).
(iii) Θ(d/ε) queries suffice for source-code access model, matching the lower bound due to Jeon and Oh (npj Quantum Inf. 2026).
This demonstrates a strict hierarchy of complexities for quantum channel certification to unitary across various access models.
Quantum process tomography, the task of estimating an unknown quantum channel, is a central problem in quantum information theory. A long-standing open question is how many uses of an unknown channel are required to learn it in diamond distance, the standard metric for distinguishing quantum processes. While quantum state tomography is well understood, for general channels the problem remained open beyond the unitary case. Here we establish the query complexity of channel tomography with optimal dependence on the dimension parameters, at any fixed constant accuracy. We design an algorithm showing that any channel with input/output dimensions din,dout and Kraus rank at most k can be learned to accuracy ε using O(dindoutk/ε2) channel uses. Conversely, we prove that Ω(dindoutk) uses are necessary at constant accuracy and that, for non-minimal Kraus rank, a separate Ω(1/ε2) contribution is unavoidable. Since channels subsume states, unitaries, isometries, and measurements as special cases, our protocol provides a unified framework for these tomography tasks, yielding new guarantees for isometry and measurement tomography while recovering known optimal scalings for state and unitary tomography. Our algorithm follows the natural strategy of performing optimal tomography on the Choi state. The main technical contribution is to show that this suffices to control the induced diamond-distance error, avoiding the dimension loss incurred by a naive conversion from Choi-state trace distance to channel diamond distance. The protocol uses the channel non-adaptively to prepare Choi-state copies, purifies them in parallel, and performs optimal pure-state tomography on the resulting purifications. Hence, we reduce channel tomography to pure-state tomography.