The complexity class Quantum Statistical Zero-Knowledge ($\mathsf{QSZK}$), introduced by Watrous (FOCS 2002) and later refined in Watrous (SICOMP, 2009), has the best known upper bound $\mathsf{QIP(2)} \cap \text{co-}\mathsf{QIP(2)}$, which was simplified following the inclusion $\mathsf{QIP(2)} \subseteq \mathsf{PSPACE}$ established in Jain, Upadhyay, and Watrous (FOCS 2009). Here, $\mathsf{QIP(2)}$ denotes the class of promise problems that admit two-message quantum interactive proof systems in which the honest prover is typically \textit{computationally unbounded}, and $\text{co-}\mathsf{QIP(2)}$ denotes the complement of $\mathsf{QIP(2)}$.
We slightly improve this upper bound to $\mathsf{QIP(2)} \cap \text{co-}\mathsf{QIP(2)}$ with a quantum linear-space honest prover. A similar improvement also applies to the upper bound for the non-interactive variant $\mathsf{NIQSZK}$. Our main techniques are an algorithmic version of the Holevo-Helstrom measurement and the Uhlmann transform, both implementable in quantum linear space, implying polynomial-time complexity in the state dimension, using the recent space-efficient quantum singular value transformation of Le Gall, Liu, and Wang (CC, to appear).
Journal reference
Proceedings of the 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026), pp. 12:1-12:20, 2026
The complexity class Quantum Statistical Zero-Knowledge (QSZK), introduced by Watrous (FOCS 2002) and later refined in Watrous (SICOMP, 2009), has the best known upper bound QIP(2)∩co-QIP(2), which was simplified following the inclusion QIP(2)⊆PSPACE established in Jain, Upadhyay, and Watrous (FOCS 2009). Here, QIP(2) denotes the class of promise problems that admit two-message quantum interactive proof systems in which the honest prover is typically \textit{computationally unbounded}, and co-QIP(2) denotes the complement of QIP(2).
We slightly improve this upper bound to QIP(2)∩co-QIP(2) with a quantum linear-space honest prover. A similar improvement also applies to the upper bound for the non-interactive variant NIQSZK. Our main techniques are an algorithmic version of the Holevo-Helstrom measurement and the Uhlmann transform, both implementable in quantum linear space, implying polynomial-time complexity in the state dimension, using the recent space-efficient quantum singular value transformation of Le Gall, Liu, and Wang (CC, to appear).
In this paper we propose a definition for (honest verifier) quantum statistical zero-knowledge interactive proof systems and study the resulting complexity class, which we denote QSZK/sub HV/. We prove several facts regarding this class, including: the following problem is a complete promise problem for QSZKHV: given instructions for preparing two mixed quantum states, are the states close together or far apart in the trace norm metric? This problem is a quantum generalization of the complete promise problem of Sahai and Vadhan (1997) for (classical) statistical zero-knowledge; QSZK/sub HV/ is closed under complement; QSZK/sub HV//spl sube/PSPACE. (At present it is not known if arbitrary quantum interactive proof systems can be simulated in PSPACE even for one-round proof systems); any polynomial-round honest verifier quantum statistical zero-knowledge proof system can be simulated by a two-message (i.e., one-round) honest verifier quantum statistical zero-knowledge proof system. Similarly, any polynomial-round honest verifier quantum statistical zero-knowledge proof system can be simulated by a three-message public-coin honest verifier quantum statistical zero-knowledge proof system. These facts establish close connections between classical statistical zero-knowledge and our definition for quantum statistical zero-knowledge, and give some insight regarding the effect of this zero-knowledge restriction on quantum interactive proof systems. The relationship between our definition and possible definitions of general (i.e., not necessarily honest) quantum statistical zero-knowledge are also discussed.
Interactive proof systems with a laconic prover, studied by Goldreich, Vadhan, and Wigderson (CC, 2002), capture problems verifiable with logarithmic prover communication in the classical setting. For two-message quantum analogs, even a single-bit prover response contains quantum statistical zero-knowledge (QSZK), introduced by Watrous (FOCS 2002). However, restricting the verifier's question to classical public coins collapses the corresponding class to BQP, as shown by Beigi, Shor, and Watrous (ToC, 2011).
We further study two-message quantum interactive proof systems with a laconic prover. To this end, we introduce the class QIPℓ-bit(2), where ℓ is the length of the prover's response, and establish:
1. A natural complete characterization of QIPℓ-bit(2) by Multi-State Distinguishability. In particular, Quantum State Distinguishability (QSD) is QIPbit-complete. Since QSD is QSZK-hard, our result places QIPℓ-bit(2), for ℓ≥2, in a landscape "just above" QSZK.
2. Easy regimes for QIPℓ-bit(2) collapsing to QSZK. We prove that QSD[a,b] (and thus QIPbit[a,b]) is in QSZK when a(n)−b(n)≥1/O(logn), and combine this with an answer compression from QIPℓ-bit[2,c,s] to QIPbit to obtain another easy regime when 2c>(1+2ℓ/2)s. Remarkably, our improved polarization applies to SD and SZK, resolving an open problem in Sahai and Vadhan (JACM, 2003).
3. Quantum public coins also make the interaction useless: qc-QAM[O(logn)] with constant gap is in BQP, where qc-QAM[ℓ] is a subclass of QIPℓ-bit(2) in which the verifier's question is exactly halves of EPR pairs.
We introduce two models of space-bounded quantum interactive proof systems, QIPL and QIP_{U}L. The QIP_{U}L model, a space-bounded variant of quantum interactive proofs (QIP) introduced by Watrous (CC 2003) and Kitaev and Watrous (STOC 2000), restricts verifier actions to unitary circuits. In contrast, QIPL allows logarithmically many pinching intermediate measurements per verifier action, making it the weakest model that encompasses the classical model of Condon and Ladner (JCSS 1995). We characterize the computational power of QIPL and QIP_{U}L. When the message number m is polynomially bounded, QIP_{U}L ⊊ QIPL unless P = NP: - QIPL^HC, a subclass of QIPL defined by a high-concentration condition on yes instances, exactly characterizes NP. - QIP_{U}L is contained in P and contains SAC¹ ∪ BQL, where SAC¹ denotes problems solvable by classical logarithmic-depth, semi-unbounded fan-in circuits. However, this distinction vanishes when m is constant. Our results further indicate that (pinching) intermediate measurements uniquely impact space-bounded quantum interactive proofs, unlike in space-bounded quantum computation, where BQL = BQ_{U}L. We also introduce space-bounded unitary quantum statistical zero-knowledge (QSZK_{U}L), a specific form of QIP_{U}L proof systems with statistical zero-knowledge against any verifier. This class is a space-bounded variant of quantum statistical zero-knowledge (QSZK) defined by Watrous (SICOMP 2009). We prove that QSZK_{U}L = BQL, implying that the statistical zero-knowledge property negates the computational advantage typically gained from the interaction.
Fidelity is a fundamental measure for the closeness of two quantum states, which is important both from a theoretical and a practical point of view. Yet, in general, it is difficult to give good estimates of fidelity, especially when one works with mixed states over Hilbert spaces of very high dimension. Although, there has been some progress on fidelity estimation, all prior work either requires a large number of identical copies of the relevant states, or relies on unproven heuristics. In this work, we improve on both of these aspects by developing new and efficient quantum algorithms for fidelity estimation with provable performance guarantees in case at least one of the states is approximately low-rank. Our algorithms use advanced quantum linear algebra techniques, such as the quantum singular value transformation, as well as density matrix exponentiation and quantum spectral sampling. As a complementary result, we prove that fidelity estimation to any non-trivial constant additive accuracy is hard in general, by giving a sample complexity lower bound that depends polynomially on the dimension. Moreover, if circuit descriptions for the relevant states are provided, we show that the task is hard for the complexity class called (honest verifier) quantum statistical zero knowledge via a reduction to a closely related result by Watrous.
Complexity theory traditionally studies the hardness of solving classical computational problems. In the quantum setting, it is also natural to consider a different notion of complexity, namely the complexity of physically preparing a certain quantum state. We study the relation between two such state complexity classes: statePSPACE, which contains states that can be generated by space-uniform polynomial-space quantum circuits, and stateQIP, which contains states that a polynomialtime quantum verifier can generate by interacting with an all-powerful untrusted quantum prover. The latter class was recently introduced by Rosenthal and Yuen (ITCS 2022), who proved that statePSPACE ⊆ stateQIP.Our main result is the reverse inclusion, stateQIP ⊆ statePSPACE, thereby establishing equality of the two classes and providing a natural state-complexity analogue to the celebrated QIP = PSPACE theorem of Jain, et al. (J. ACM 2011). To prove this, we develop a polynomial-space quantum algorithm for solving a large class of exponentially large “PSPACE-computable” semidefinite programs (SDPs), which also prepares an optimiser encoded in a quantum state. Our SDP solver relies on recent blockencoding techniques from quantum algorithms, demonstrating that these techniques are also useful for complexity theory.Using similar techniques, we also show that optimal prover strategies for general quantum interactive protocols can be implemented in quantum polynomial space. We prove this by studying an algorithmic version of Uhlmann’s theorem and establishing an upper bound on the complexity of implementing Uhlmann transformations.
Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations. Our QMA proof system is sound against arbitrary quantum provers, but only requires an honest prover to perform polynomial-time quantum computations, provided that it holds a quantum witness for a given instance of the QMA problem under consideration. The proof system relies on a new variant of the QMA-complete local Hamiltonian problem in which the local terms are described by Clifford operations and standard basis measurements. We believe that the QMA-completeness of this problem may have other uses in quantum complexity.