Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2510.20789

The complexity of perfect quantum state classification

Nathaniel Johnston, Benjamin Lovitz, Russo, Vincent, Jamie Sikora

Published Oct 23, 2025• 0 citations
arXivHTMLPDF

Abstract

The problem of quantum state classification asks how accurately one can identify an unknown quantum state that is promised to be drawn from a known set of pure states. In this work, we introduce the notion of $k$-learnability, which captures the ability to identify the correct state using at most $k$ guesses, with zero error. We show that deciding whether a given family of states is $k$-learnable can be solved via semidefinite programming. When there are $n$ states, we present polynomial-time (in $n$) algorithms for determining $k$-learnability for two cases: when $k$ is a fixed constant or the dimension of the states is a fixed constant. When both $k$ and the dimension of the states are part of the input, we prove that there exist succinct certificates placing the problem in NP, and we establish NP-hardness by a reduction from the classical $k$-clique problem. Together, our findings delineate the boundary between efficiently solvable and intractable instances of quantum state classification in the perfect (zero-error) regime.

DOI
10.48550/arxiv.2510.20789

Related papers

Sharp bounds for perfect quantum state classification beyond antidistinguishability

Nathaniel Johnston, Benjamin Lovitz, Vincent Russo +1 more

A multiset of pure quantum states is said to be k-learnable if there is a measurement strategy that always narrows an unknown sample drawn from the list down to one of at most k candidates. The parameter k interpolates between distinguishability and antidistinguishability, and provides a unified framework for partial state identification. We prove two universal, and optimal, Gram-matrix criteria for k-learnability: a Frobenius-norm sufficient condition and an entrywise-$\ell_1$ necessary condition. We apply them to derive explicit learnability and copy-complexity guarantees for several well-known sets of states including SIC-POVMs, mutually unbiased bases, and stabilizer states. We further apply our results to zero-error mutation detection problems such as anomaly detection and changepoint detection.

quant-pharXiv:2609.17411•Sep 16, 2026•0 citations

On the Complexity of Computing Zero-Error and Holevo Capacity of Quantum Channels

Salman Beigi, Peter W. Shor

One of the main problems in quantum complexity theory is that our understanding of the theory of QMA-completeness is not as rich as its classical analogue, the NP- completeness. In this paper we consider the clique problem in graphs, which is NP- complete, and try to find its quantum analogue. We show that, quantum clique problem can be defined as follows; Given a quantum channel, decide whether there are k states that are distinguishable, with no error, after passing through channel. This definition comes from reconsidering the clique problem in terms of the zero-error capacity of graphs, and then redefining it in quantum information theory. We prove that, quantum clique problem is QMA-complete. In the second part of paper, we consider the same problem for the Holevo capacity. We prove that computing the Holevo capacity as well as the minimum entropy of a quantum channel is NP-complete. Also, we show these results hold even if the set of quantum channels is restricted to entanglement breaking ones.

arXiv:0709.2090•Sep 13, 2007•16 citations

Computational Complexity of Learning Efficiently Generatable Pure States

Taiga Hiroka, Min-Hsiu Hsieh

Understanding the computational complexity of learning efficient classical programs in various learning models has been a fundamental and important question in classical computational learning theory. In this work, we study the computational complexity of quantum state learning, which can be seen as a quantum generalization of distributional learning introduced by Kearns et.al [STOC94]. Previous works by Chung and Lin [TQC21], and Bădescu and O$'$Donnell [STOC21] study the sample complexity of the quantum state learning and show that polynomial copies are sufficient if unknown quantum states are promised efficiently generatable. However, their algorithms are inefficient, and the computational complexity of this learning problem remains unresolved. In this work, we study the computational complexity of quantum state learning when the states are promised to be efficiently generatable. We show that if unknown quantum states are promised to be pure states and efficiently generateable, then there exists a quantum polynomial time algorithm $A$ and a language $L \in PP$ such that $A^L$ can learn its classical description. We also observe the connection between the hardness of learning quantum states and quantum cryptography. We show that the existence of one-way state generators with pure state outputs is equivalent to the average-case hardness of learning pure states. Additionally, we show that the existence of EFI implies the average-case hardness of learning mixed states.

arXiv:2410.04373•Oct 6, 2024•0 citations

On the Complexity of Computing Zero-Error and Holevo Capacity of Quantum Channels

Salman BeigiPeter, W. Shor

One of the main problems in quantum complexity theory is that our understanding of the theory of QMA-completeness is not as rich as its classical analogue, the NPcompleteness. In this paper we consider the clique problem in graphs, which is NPcomplete, and try to find its quantum analogue. We show that, quantum clique problem can be defined as follows; Given a quantum channel, decide whether there are k states that are distinguishable, with no error, after passing through channel. This definition comes from reconsidering the clique problem in terms of the zero-error capacity of graphs, and then redefining it in quantum information theory. We prove that, quantum clique problem is QMA-complete. In the second part of paper, we consider the same problem for the Holevo capacity. We prove that computing the Holevo capacity as well as the minimum entropy of a quantum channel is NP-complete. Also, we show these results hold even if the set of quantum channels is restricted to entanglement breaking ones.

•Jan 1, 2008•59 citations

Quantum Hardness of Learning Shallow Classical Circuits

Srinivasan Arunachalam, Alex B. Grilo, Aarthi Sundaram

In this paper, we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of probably approximately correct (PAC) learning. In order to attain our results, we establish connections between quantum learning and quantum-secure cryptosystems. We then achieve the following results. 1. Hardness of PAC learning ${AC}^0$ and ${TC}^0$ under the uniform distribution. Our first result concerns the concept class ${TC}^0$ (resp., ${AC}^0$), the class of constant-depth, polynomial-sized circuits with unbounded fan-in majority gates (resp., ${AND}, {OR}, {NOT}$ gates). We show the following: if there exists no quantum (quasi-)polynomial-time algorithm to solve the ring-learning with errors (${RLWE}$) problem, then there exists no (quasi-)polynomial-time quantum learning algorithm for ${TC}^0$; and if there exists no $2^{O(d^{1/\eta})}$-time quantum algorithm to solve ${RLWE}$ with dimension $d = O(polylog n)$ (for every constant $\eta > 2$), then there exists no $O(n^{ \log^{\nu} n} )$-time quantum learning algorithm for $poly(n)$-sized ${AC}^0$ circuits (for a constant $\nu>0$), matching the classical upper bound of Linial, Mansour and Nisan [J. ACM, 40 (1993), pp. 607--620], where the learning algorithms are under the uniform distribution (even with access to quantum membership queries). The main technique in these results uses an explicit family of pseudorandom functions that are believed to be quantum-secure to construct concept classes that are hard to learn quantumly under the uniform distribution. 2. Hardness of learning ${TC}^0_2$ in the PAC setting. Our second result shows that if there exists no quantum polynomial-time algorithm for the ${LWE}$ problem, then there exists no polynomial-time quantum-PAC learning algorithm for the class ${TC}^0_2$, i.e., depth-2 ${TC}^0$ circuits. The main technique in this result is to establish a connection between the quantum security of public-key encryption schemes and the learnability of a concept class that consists of decryption functions of the cryptosystem. Our results show that quantum resources do not give an exponential improvement to learning constant-depth polynomial-sized neural networks. This also gives a strong (conditional) negative answer to one of the “Ten Semi-Grand Challenges for Quantum Computing Theory" raised by Aaronson https://www.scottaaronson.com/writings/qchallenge.html, 2005.

arXiv:1903.02840•SIAM Journal on Computing•Jan 1, 2021•11 citations

Learning the Closest Product State

Ainesh Bakshi, John Bostanci, William Kretschmer +5 more

We study the problem of finding a (pure) product state with optimal fidelity to an unknown $n$-qubit quantum state $ρ$, given copies of $ρ$. This is a basic instance of a fundamental question in quantum learning: is it possible to efficiently learn a simple approximation to an arbitrary state? We give an algorithm which finds a product state with fidelity $\varepsilon$-close to optimal, using $N = n^{\text{poly}(1/\varepsilon)}$ copies of $ρ$ and $\text{poly}(N)$ classical overhead. We further show that estimating the optimal fidelity is NP-hard for error $\varepsilon = 1/\text{poly}(n)$, showing that the error dependence cannot be significantly improved. For our algorithm, we build a carefully-defined cover over candidate product states, qubit by qubit, and then demonstrate that extending the cover can be reduced to approximate constrained polynomial optimization. For our proof of hardness, we give a formal reduction from polynomial optimization to finding the closest product state. Together, these results demonstrate a fundamental connection between these two seemingly unrelated questions. Building on our general approach, we also develop more efficient algorithms in three simpler settings: when the optimal fidelity exceeds $5/6$; when we restrict ourselves to a discrete class of product states; and when we are allowed to output a matrix product state.

arXiv:2411.04283•ACM Symposium on Theory of Computing (STOC)•Jun 15, 2025•1 citation
The problem of quantum state classification asks how accurately one can identify an unknown quantum state that is promised to be drawn from a known set of pure states. In this work, we introduce the notion of kkk-learnability, which captures the ability to identify the correct state using at most kkk guesses, with zero error. We show that deciding whether a given family of states is kkk-learnable can be solved via semidefinite programming. When there are nnn states, we present polynomial-time (in nnn) algorithms for determining kkk-learnability for two cases: when kkk is a fixed constant or the dimension of the states is a fixed constant. When both kkk and the dimension of the states are part of the input, we prove that there exist succinct certificates placing the problem in NP, and we establish NP-hardness by a reduction from the classical kkk-clique problem. Together, our findings delineate the boundary between efficiently solvable and intractable instances of quantum state classification in the perfect (zero-error) regime.
A multiset of pure quantum states is said to be k-learnable if there is a measurement strategy that always narrows an unknown sample drawn from the list down to one of at most k candidates. The parameter k interpolates between distinguishability and antidistinguishability, and provides a unified framework for partial state identification. We prove two universal, and optimal, Gram-matrix criteria for k-learnability: a Frobenius-norm sufficient condition and an entrywise-ℓ1\ell_1ℓ1​ necessary condition. We apply them to derive explicit learnability and copy-complexity guarantees for several well-known sets of states including SIC-POVMs, mutually unbiased bases, and stabilizer states. We further apply our results to zero-error mutation detection problems such as anomaly detection and changepoint detection.
One of the main problems in quantum complexity theory is that our understanding of the theory of QMA-completeness is not as rich as its classical analogue, the NP- completeness. In this paper we consider the clique problem in graphs, which is NP- complete, and try to find its quantum analogue. We show that, quantum clique problem can be defined as follows; Given a quantum channel, decide whether there are k states that are distinguishable, with no error, after passing through channel. This definition comes from reconsidering the clique problem in terms of the zero-error capacity of graphs, and then redefining it in quantum information theory. We prove that, quantum clique problem is QMA-complete. In the second part of paper, we consider the same problem for the Holevo capacity. We prove that computing the Holevo capacity as well as the minimum entropy of a quantum channel is NP-complete. Also, we show these results hold even if the set of quantum channels is restricted to entanglement breaking ones.
Understanding the computational complexity of learning efficient classical programs in various learning models has been a fundamental and important question in classical computational learning theory. In this work, we study the computational complexity of quantum state learning, which can be seen as a quantum generalization of distributional learning introduced by Kearns et.al [STOC94]. Previous works by Chung and Lin [TQC21], and Bădescu and O′'′Donnell [STOC21] study the sample complexity of the quantum state learning and show that polynomial copies are sufficient if unknown quantum states are promised efficiently generatable. However, their algorithms are inefficient, and the computational complexity of this learning problem remains unresolved. In this work, we study the computational complexity of quantum state learning when the states are promised to be efficiently generatable. We show that if unknown quantum states are promised to be pure states and efficiently generateable, then there exists a quantum polynomial time algorithm AAA and a language L∈PPL \in PPL∈PP such that ALA^LAL can learn its classical description. We also observe the connection between the hardness of learning quantum states and quantum cryptography. We show that the existence of one-way state generators with pure state outputs is equivalent to the average-case hardness of learning pure states. Additionally, we show that the existence of EFI implies the average-case hardness of learning mixed states.
One of the main problems in quantum complexity theory is that our understanding of the theory of QMA-completeness is not as rich as its classical analogue, the NPcompleteness. In this paper we consider the clique problem in graphs, which is NPcomplete, and try to find its quantum analogue. We show that, quantum clique problem can be defined as follows; Given a quantum channel, decide whether there are k states that are distinguishable, with no error, after passing through channel. This definition comes from reconsidering the clique problem in terms of the zero-error capacity of graphs, and then redefining it in quantum information theory. We prove that, quantum clique problem is QMA-complete. In the second part of paper, we consider the same problem for the Holevo capacity. We prove that computing the Holevo capacity as well as the minimum entropy of a quantum channel is NP-complete. Also, we show these results hold even if the set of quantum channels is restricted to entanglement breaking ones.
In this paper, we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of probably approximately correct (PAC) learning. In order to attain our results, we establish connections between quantum learning and quantum-secure cryptosystems. We then achieve the following results. 1. Hardness of PAC learning AC0{AC}^0AC0 and TC0{TC}^0TC0 under the uniform distribution. Our first result concerns the concept class TC0{TC}^0TC0 (resp., AC0{AC}^0AC0), the class of constant-depth, polynomial-sized circuits with unbounded fan-in majority gates (resp., AND,OR,NOT{AND}, {OR}, {NOT}AND,OR,NOT gates). We show the following: if there exists no quantum (quasi-)polynomial-time algorithm to solve the ring-learning with errors (RLWE{RLWE}RLWE) problem, then there exists no (quasi-)polynomial-time quantum learning algorithm for TC0{TC}^0TC0; and if there exists no 2O(d1/η)2^{O(d^{1/\eta})}2O(d1/η)-time quantum algorithm to solve RLWE{RLWE}RLWE with dimension d=O(polylogn)d = O(polylog n)d=O(polylogn) (for every constant η>2\eta > 2η>2), then there exists no O(nlog⁡νn)O(n^{ \log^{\nu} n} )O(nlogνn)-time quantum learning algorithm for poly(n)poly(n)poly(n)-sized AC0{AC}^0AC0 circuits (for a constant ν>0\nu>0ν>0), matching the classical upper bound of Linial, Mansour and Nisan [J. ACM, 40 (1993), pp. 607--620], where the learning algorithms are under the uniform distribution (even with access to quantum membership queries). The main technique in these results uses an explicit family of pseudorandom functions that are believed to be quantum-secure to construct concept classes that are hard to learn quantumly under the uniform distribution. 2. Hardness of learning TC20{TC}^0_2TC20​ in the PAC setting. Our second result shows that if there exists no quantum polynomial-time algorithm for the LWE{LWE}LWE problem, then there exists no polynomial-time quantum-PAC learning algorithm for the class TC20{TC}^0_2TC20​, i.e., depth-2 TC0{TC}^0TC0 circuits. The main technique in this result is to establish a connection between the quantum security of public-key encryption schemes and the learnability of a concept class that consists of decryption functions of the cryptosystem. Our results show that quantum resources do not give an exponential improvement to learning constant-depth polynomial-sized neural networks. This also gives a strong (conditional) negative answer to one of the “Ten Semi-Grand Challenges for Quantum Computing Theory" raised by Aaronson https://www.scottaaronson.com/writings/qchallenge.html, 2005.
We study the problem of finding a (pure) product state with optimal fidelity to an unknown nnn-qubit quantum state ρρρ, given copies of ρρρ. This is a basic instance of a fundamental question in quantum learning: is it possible to efficiently learn a simple approximation to an arbitrary state? We give an algorithm which finds a product state with fidelity ε\varepsilonε-close to optimal, using N=npoly(1/ε)N = n^{\text{poly}(1/\varepsilon)}N=npoly(1/ε) copies of ρρρ and poly(N)\text{poly}(N)poly(N) classical overhead. We further show that estimating the optimal fidelity is NP-hard for error ε=1/poly(n)\varepsilon = 1/\text{poly}(n)ε=1/poly(n), showing that the error dependence cannot be significantly improved. For our algorithm, we build a carefully-defined cover over candidate product states, qubit by qubit, and then demonstrate that extending the cover can be reduced to approximate constrained polynomial optimization. For our proof of hardness, we give a formal reduction from polynomial optimization to finding the closest product state. Together, these results demonstrate a fundamental connection between these two seemingly unrelated questions. Building on our general approach, we also develop more efficient algorithms in three simpler settings: when the optimal fidelity exceeds 5/65/65/6; when we restrict ourselves to a discrete class of product states; and when we are allowed to output a matrix product state.