Title: Purest Quantum State Identification
Venue: NeurIPS 2025
Accepted: yes
Keywords: quantum property learning

best arm identification

online learning
Abstract: Quantum noise constitutes a fundamental obstacle to realizing practical quantum technologies. To address the pivotal challenge of identifying quantum systems least affected by noise, we introduce the purest quantum state identification, which can be used to improve the accuracy of quantum computation and communication. We formulate a rigorous paradigm for identifying the purest quantum state among $K$ unknown $n$-qubit quantum states using total $N$ quantum state copies. For incoherent strategies, we derive the first adaptive algorithm achieving error probability $\exp\left(- \Omega\left(\frac{N H_1}{\log(K) 2^n }\right) \right)$, fundamentally improving quantum property learning through measurement optimization. By developing a coherent measurement protocol with error bound $\exp\left(- \Omega\left(\frac{N H_2}{\log(K) }\right) \right)$, we demonstrate a significant separation from incoherent strategies, formally quantifying the power of quantum memory and coherent measurement. Furthermore, we establish a lower bound by demonstrating that all strategies with fixed two-outcome incoherent POVM must suffer error probability exceeding $ \exp\left( - O\left(\frac{NH_1}{2^n}\right)\right)$. This research advances the characterization of quantum noise through efficient learning frameworks. Our results establish theoretical foundations for noise-adaptive quantum property learning while delivering practical protocols for enhancing the reliability of quantum hardware.

Purest Quantum State Identification

Abstract

Quantum noise constitutes a fundamental obstacle to realizing practical quantum technologies. To address the pivotal challenge of identifying quantum systems least affected by noise, we introduce the purest quantum state identification, which can be used to improve the accuracy of quantum computation and communication. We formulate a rigorous paradigm for identifying the purest quantum state among K unknown n -qubit quantum states using total N quantum state copies. For incoherent strategies, we derive the first adaptive algorithm achieving error probability exp ( -Ω ( NH 1 log( K )2 n )) , fundamentally improving quantum property learning through measurement optimization. By developing a coherent measurement protocol with error bound exp ( -Ω ( NH 2 log( K ) )) , we demonstrate a significant separation from incoherent strategies, formally quantifying the power of quantum memory and coherent measurement. Furthermore, we establish a lower bound by demonstrating that all strategies with fixed two-outcome incoherent POVM must suffer error probability exceeding exp ( -O ( NH 1 2 n )) . This research advances the characterization of quantum noise through efficient learning frameworks. Our results establish theoretical foundations for noise-adaptive quantum property learning while delivering practical protocols for enhancing the reliability of quantum hardware.

1 Introduction

Quantum computing demonstrates the remarkable potential for solving computational problems that are intractable for classical systems [1, 2]. However, current quantum systems face two fundamental constraints that limit their practical applications: (1) the number of available quantum bits (qubits) in existing devices remains severely limited compared to theoretical requirements [3, 4], and (2) these qubits exhibit high sensitivity to environmental noise, compromising the reliability of quantum operations [5]. These fundamental constraints impose substantial barriers to achieving reliable quantum control and measurement, prerequisites for unlocking practical quantum computational advantages. Owing to variations in device implementation and environmental factors, the performance of quantum computation and communication differs markedly among various quantum systems [6]. Thus, identifying the optimal quantum device [5], quantum state, or quantum channel [7] is significant for the near-term quantum science and technology.

To evaluate the quality of quantum systems, researchers often use the purity of quantum states as a criterion. In quantum communication, high-accuracy quantum channels are critical to ensuring the accurate transmission of quantum information [7]. By measuring the purity of the channel's

associated state (Choi-state [7]), researchers can evaluate how noise affects the channel. On the other hand, preparing high-purity quantum states is also the foundation for quantum algorithms. The majority of quantum algorithms rely on pure quantum states as inputs [1, 8, 9]. Therefore, identifying high-purity quantum states is essential for accurate quantum computing.

In this paper, we study the problem of purest quantum state identification (PQSI). The learner aims to identify the purest state among multiple unknown quantum states using N copies across all states. This issue holds considerable importance from both theoretical and practical perspectives. By identifying the purest quantum state, it is achievable to distinguish the best quantum system that is least affected by noise, thus improving the accuracy of quantum computing and quantum communication in near-term quantum systems.

This paper makes the following key contributions:

Problem Model. To the best of our knowledge, this is the first study dedicated to the best quantum state identification. In this paper, we focus on the issue of the purest quantum state identification, i.e., identifying the purest quantum state from a set of available quantum states. In this problem, while allocating copies to different quantum systems, the learner also needs to select the basis for quantum measurement, which significantly increases its complexity. Due to the limited number of qubits in existing quantum devices, these systems may not be able to facilitate large-scale quantum measurement [10, 11]. Consequently, while coherent (multi-copy) measurement techniques can more efficiently acquire information about quantum states, research on incoherent (single-copy) measurement is more practical. We formalize the problem of purest quantum state identification with incoherent (single-copy) measurement as follows:

Problem 1.1 (Purest quantum state identification (PQSI) with incoherent (single-copy) measurement) . There is a set of K unknown n -qubit quantum states represented as S = { ρ 1 , . . . , ρ K } . The learner aims to identify the purest quantum state in S using a total of N copies across all states. The problem protocol at round t ∈ { 1 , ..., N } is as follows:

· The learner chooses a quantum state σ t from the set S and gets a copy of it.

· The learner selects a POVM and uses it to measure σ t , after which σ t is destroyed.

Upon completing N measurements, the learner selects a quantum state ρ ′ ∈ S based on the measurement outcomes. The objective of the learner is to maximize the probability of identifying the purest quantum state, i.e., P ( ρ ′ ∈ arg max ρ ∈ S Tr( ρ 2 )) .

In general, when coherent measurements are available, we formalize the problem as follows:

Problem 1.2 (Purest quantum state identification(PQSI) with coherent (multi-copy) measurement) . There is a set of K unknown n -qubit quantum states represented as S = { ρ 1 , . . . , ρ K } . The learner aims to identify the purest quantum state in S using a total of N copies across all states. Let N t denote the number of quantum state copies that remain available in round t and N 1 = N . When N t > 0 , the problem protocol at round t ∈ N + is as follows:

· The learner decides the number of copies of the quantum state ρ i to acquire, denoted as s ( i, t ) , where s ( i, t ) ≥ 0 and 1 ≤ ∑ K i =1 s ( i, t ) ≤ N t . The quantum state constructed from these copies can be represented as σ t = ρ ⊗ s (1 ,t ) 1 ⊗ . . . ⊗ ρ ⊗ s ( K,t ) K . Let N t +1 = N t -∑ K i =1 s ( i, t ) .

· The learner selects an entangled POVM and uses it to measure σ t , after which σ t is destroyed.

Upon completing measurements of all N copies of quantum states, the learner selects a quantum state ρ ′ ∈ S based on the measurement outcomes. The objective of the learner is to maximize the probability of identifying the purest quantum state P ( ρ ′ ∈ arg max ρ ∈ S Tr( ρ 2 )) .

Algorithms. We develop two distinct algorithms to address this problem in different settings. To simplify the expression, for i ∈ { 1 , ..., K } , let ρ ( i ) be the i -th purest quantum state in S and ∆ ( i ) = Tr ( ρ 2 (1) ) -Tr ( ρ 2 ( i ) ) be the purity gap between the optimal quantum state and the i -th purest quantum state. In scenarios where only single-copy measurements are available, we developed the incoherent measurement based algorithm IM-PQSI. We employ Haar unitary matrices to generate random measurement basis and to measure quantum states in S . This measurement approach allows us to evaluate the purity of quantum states by analyzing measurement expectations and variances. The Haar unitary matrices utilized in IM-PQSI belong to the class of unitary 4-designs,

which can be efficiently simulated on a quantum computer up to inverse-exponential trace distance [12, 13, 14, 15, 16]. During the evaluation of quantum state purity, our algorithms dynamically allocate copies to various quantum states. By assigning more copies to states with higher purity, we enhance the estimation accuracy of these states, thereby increasing the likelihood of identifying the purest quantum state. Furthermore, we improve the performance of the algorithm for identifying the optimal quantum state by balancing the selection of measurement basis with the number of measurements conducted for each base. The error probability of this algorithm satisfies the following theorem:

Theorem 1.3 (Informal version of Theorem 4.1) . There exists an algorithm that solves the problem of the purest quantum state identification with incoherent measurement whose error probability satisfies

where H 1 = min i ∈{ 2 ,...,K } ∆ ( i )

Conversely, when multi-copy measurements are accessible, we developed the coherent measurement based algorithm CM-PQSI. We use the SWAP test to estimate the purity of quantum states in this algorithm, and its error probability satisfies the following theorem:

Theorem 1.4 (Informal version of Theorem 6.1) . There exists an algorithm that solves the problem of the purest quantum state identification with coherent measurement whose error probability satisfies

where H 2 = min i ∈{ 2 ,...,K } ∆ 2 ( i ) i .

By comparing the error probability upper bound of these two algorithms, we can identify the advantages of quantum memory.

Lower Bound. For incoherent measurements, we utilize the properties of Haar unitary matrices to formulate a related problem named Purest Random Quantum State Identification (PRQSI). In this context, the learner is required to consider a specific quantum state distribution constructed from Haar unitary matrices. We demonstrate that the lower bound of the error probability for the PRQSI problem is also applicable as the lower bound for the error probability of the PQSI problem. Furthermore, we show that the measurement outcomes generated by three-fourths of the Haar unitary matrices are insufficient for the learner to easily differentiate among them, presenting a challenge in identification. Additionally, we demonstrate that in any PRQSI problem employing a fixed two-outcome Positive Operator-Valued Measure, the measurement outcomes exhibit a Bernoulli distribution that is inherently difficult to distinguish. Consequently, we derive the lower bounds for the error probability of the PRQSI, which are also the lower bounds for the error probability of the PQSI problem, as follows:

Theorem 1.5 (Informal version of Theorem 5.6) . For any algorithm A to solve the purest quantum state identification using fixed 2-outcome randomly incoherent POVM, there exists a set of quantum states which makes the error probability of A satisfies

where H 1 = min i ∈{ 2 ,...,K } ∆ ( i ) i .

2 Related Work

Quantum learning and testing. Quantum learning and testing [17, 18] is a vital area of research in quantum computing and quantum communication. Extensive investigations have been conducted to understand the complexities of various measurements. Quantum state tomography [19, 20, 21, 22] involves obtaining complete information about the density matrix of a quantum state through measurements. While this technique can be employed to tackle the PQSI problem, it incurs significant sampling costs. For the quantum state certification [23, 24, 25, 26], the target is to determine whether

a quantum state is close to a specific target quantum state. Our problem can be viewed as identifying the quantum state that is the farthest from the maximally mixed state. However, this problem is focused on quantum testing and does not deal with distance estimation.

Hence, these methods are not applicable to the PQSI problem. Another category of problems relates to inner product estimation between two quantum states [27, 28, 29, 30]. When proving lower bounds, this category often significantly restricts the quantum states for distinction. The relevant literature employs two general approaches to establish problem complexity. The first approach involves constructing counterexamples using Haar unitary matrices [27, 22, 26, 23]. However, the representation-theoretic structure of Haar unitary matrices is intricate [31], which makes it difficult to use. The other approach uses Gaussian Orthogonal Ensemble matrices to create counterexamples [24, 21]. However, when using GOE to prove the lower bound, the distance between quantum states is in a specific range rather than a fixed number, making it unsuitable for the PQSI problem.

Classical best arm identification. To the best of our knowledge, our work is the first to consider the best quantum state identification. Among the classical learning tasks, the best arm identification [32, 33, 34, 35, 36] has been extensively studied and is divided into two categories: fixed budget [37] and fixed confidence [38]. However, the existing research can only deal with the problem under specific distributions. These limitations restrict the algorithm's applicability and leave considerable room for further research on this issue. In our problem, we must select an appropriate POVM basis while choosing the quantum state in each round. The quantum state space and the POVM space grow exponentially with the increase in qubits, which makes this problem significantly more challenging than solving a classical problem of the best arm identification.

3 Preliminaries and Notations

Quantum State. Let d = 2 n denote the dimension of an n -qubit quantum system. An n -qubit quantum state can be represented by a density matrix ρ ∈ C d × d , which is Hermitian and trace-1 positive semi-definite. In particular, an n -qubit pure quantum state can be represented by a unit vector | ψ ⟩ ∈ C d . The purity of a quantum state ρ is Tr( ρ 2 ) .

Quantum Measurement. Quantum measurements are usually described by a Positive OperatorValued Measure (POVM), which produces probabilistic outcomes. An n -qubit positive operatorvalued measurement M can be represented as a collection of positive semi-definite matrices M = { M m } m , where M m ∈ C d × d and ∑ m M m = I d . When using M to measure a quantum state ρ , the probability of outcome m is Tr( M m ρ ) , and the quantum state ρ is destroyed. When the coherent measurement method is employed, the learner can perform entanglement measurements on quantum states ρ 1 ⊗ . . . ⊗ ρ m . However, this approach necessitates the support of large-scale quantum devices and quantum memory, which are not feasible with current quantum technologies. Therefore, researching incoherent measurement methods applicable to near-term quantum devices is of great significance.

̸

In this study, we aim to identify the purest quantum state from a set of unknown quantum states, achieving the highest probability through N measurements. For the purpose of simplicity, we will assume that there exists a unique optimal quantum state that is the purest in the set S , denoted as µ ⋆ = µ i ⋆ . For i = i ⋆ , we represent the purity difference between each non-optimal quantum state and the optimal quantum state using

For i ∈ { 1 , ..., K } , let ρ ( i ) be the i -th purest quantum state in S , then we have

and

Let e N denote the probability that the learner does not choose the purest quantum state in S after N samples and measurements, i.e., e N = P ( ρ ′ / ∈ arg max ρ Tr( ρ 2 ) ) . The learner's objective is to min e N . We summarize key notations used throughout this paper in Table 1.

4 Algorithm for PQSI with incoherent measurement

The current era of quantum computing presents limitations in the number of available qubits. This may hinder the measurement of multiple quantum states operated jointly. In this section, we use incoherent (single-copy) measurement methods in each round to select the purest quantum state from the set of quantum states.

The algorithm we designed to solve the PQSI problem with incoherent measurements is shown in Algorithm 1. In our algorithm, we utilize Haar-random unitary matrices to construct multiple random measurement bases for probing quantum state copies. For each constructed measurement basis, we perform m repeated measurements on the same quantum state. Let the i -th measurement outcome be x i ∈ { 0 , . . . , d -1 } for a d -outcome POVM. We estimate the quantum state's purity based on the empirical probability of obtaining identical outcomes, defined as

̸

which corresponds to the well-known collision estimator [39, 40] used to estimate the second moment ∑ d i =1 p 2 i in classical statistics.

The rationale behind this choice is as follows. The positive operator-valued measurements (POVMs) adopted in our algorithm have d possible outcomes, where p i denotes the probability of observing the i -th outcome. The purity of a quantum state is closely related to the quantity ∑ d i =1 p 2 i . Hence, an essential step in our algorithm is to estimate this second-order statistic accurately. The collision estimator provides a natural and statistically efficient approach to this task, allowing us to connect the observed measurement coincidences to the underlying purity of the quantum state. By analyzing the expectation and variance of ˜ g , we can further quantify the probability of correctly identifying the purest quantum state within our framework.

Algorithm 1 includes two random processes. The first process involves the random selection of Haar unitary matrices to construct measurement bases, while the second process entails the random acquisition of measurement results using these bases. Balancing the estimation errors introduced by these two processes is crucial for enhancing the algorithm's accuracy. When the purity difference between quantum states is substantial, each measurement base requires fewer instances to distinguish purity differences among states effectively. Conversely, when the purity difference is minimal, more measurements are necessary under a single measurement base to gather sufficient information, limiting the number of measurement bases that can be utilized. Furthermore, when the purity difference is extremely slight, the choice of measurement basis significantly influences the measurement outcomes. Performing Θ( d 2 ) measurements under a single measurement base is essential for adequately differentiating the outcomes between quantum states. Additionally, increasing the number of measurements under one measurement base will not improve the accuracy of purity estimation. Given these considerations, we balance the number of measurement bases and the number of measurements per base to enhance the probability of identifying the optimal quantum state.

In Algorithm 1, we use Haar unitary matrices to construct random measurement bases. Haar unitary matrices are extensively employed in quantum property testing and learning theory [41, 27, 30]. Achieving exact Haar randomness is challenging; therefore, unitary t -designs are typically used to approximate Haar unitary matrices. Haar unitary matrices used in Algorithm 1 can be approximated using unitary 4 -designs. As established in [12], unitary t -designs can be efficiently implemented on quantum computers with exponentially minor approximation errors. This theoretical foundation ensures the practical feasibility and mathematical validity of employing Haar unitary matrices in our algorithmic framework.

Then, we give the error probability upper bound of IM-PQSI in the following theorem.

Theorem 4.1. For i ∈ { 1 , ..., K } , ∆ i ≥ c > 1 d 2 , where c is a constant. Set m = ⌈ 1 √ c ⌉ in Algorithm 1. The error probability of Algorithm 1 satisfies

where H 1 = min i ∈{ 2 ,...,K } ∆ ( i ) i .

Algorithm 1 Incoherent measurement based algorithm for solving PQSI problem (IM-PQSI)

Input: Copy access to S = { ρ 1 , ..., ρ K } , sample number N .

Initialization:

Set

S

0

=

{

ρ

1

, ..., ρ

K

}

,

log(

K

) =

+

∑

K

i

=2

,

N

0

= 0

and

N

k

=

⌈ 1 log( K ) N -K K +1 -k ⌉ , for k ∈ { 1 , ..., K -1 } . Sample ⌊ N/m ⌋ random unitary matrix U 1 , ..., U ⌊ N/m ⌋ according to the Haar measure.

for k=1,..., K-1 do

Measure m copies of ρ in the basis { U † j | i ⟩⟨ i | U j } d -1 i =0 and set the outputs as x ( ρ, j, 1) , ..., x ( ρ, j, m ) .

Let S k = S k -1 \ arg min ρ ∈ S k -1 w ( ρ, k ) . end for

Output the quantum state ρ ′ in S k -1 .

Proof Sketch. The expectation of w ( ρ, k ) and ˜ g ρ,j in Algorithm 1 satisfies E [ w ( ρ, k )] = E [˜ g ρ,j ] = ( m -1)(1+Tr( ρ 2 )) m ( d +1) , and the variance of ˜ g ( ρ, j ) satisfies Var(˜ g ( ρ, j )) = O ( 1 d 3 + 1 m 2 d + 1 md 2 ) . By Bernstein's inequality and the union bound of error probability, we have e n ≤ K ( K -1) 2 exp ( -Ω ( √ cNH 1 log( K ) d )) where H 1 = min i ∈{ 1 ,...,K } ∆ ( i ) i . The proof details are provided in Appendix B.2.

When the purity gaps between quantum states are very small, the gap parameter H 1 becomes correspondingly small, which naturally results in a looser error bound. This reflects the intrinsic difficulty of the problem: identifying the quantum state with the highest purity is fundamentally hard when multiple states have nearly identical purities. Moreover, the system dimension d = 2 n grows exponentially with the number of qubits n , so that 1 d 2 tends to zero as n increases. Therefore, in Theorem 4.1, we assume that for any quantum state ρ ∈ S ρ , Tr( ρ ⋆ 2 ) -Tr( ρ 2 ) ≥ c > 1 d 2 . If this assumption does not hold, we can instead derive the following conclusion:

Lemma 4.2. Set m = d in Algorithm 1. The probability of error of Algorithm 1 satisfies

where H 1 = min i ∈{ 2 ,...,K } ∆ ( i ) i , and H 2 = min i ∈{ 2 ,...,K } ∆ 2 ( i ) i .

Appendix B.3 provides the proof details of Lemma 4.2.

In the field of quantum learning and testing, research on quantum channels constitutes a critical aspect. Evaluating the impact of noise on quantum channels can significantly enhance the accuracy of quantum computing and quantum communication [7]. Introduced a method for assessing the 'unitarity" of a quantum channel by evaluating the purity of a quantum state. Subsequently, we can utilize the algorithm IM-PQSI to identify the most 'unitary" quantum channel from a quantum channel set. Let u ( i ) denote the unitarity of the i -th most unitary quantum channel. We have the following corollary:

Corollary 4.3. There exists an algorithm that solves the problem of the most 'unitary" channel identification with incoherent access whose error probability satisfies:

where H u = min i ∈{ 2 ,...,K } u (1) -u ( i ) i .

1

2

1

i

5 Lower bound for PQSI with incoherent measurement

In this section, we investigate the lower bound on the error probability for solving the problem of purest quantum state identification. This problem requires distinguishing between quantum states with different purity through sampling and measurement. Recent studies [31] indicated that when a quantum state ρ is rotated by a Haar unitary matrix and measured N times, the output distribution can be calculated only if ρ is either a pure state or a maximally mixed state. Consequently, the complexity analysis of testing problems often assumes that one of the quantum states is either a pure state or a maximally mixed state. This limitation presents significant challenges to our analysis.

To solve this problem, we reduce the problem of identifying the purest quantum state to the problem of identifying the purest random quantum state. This reduction allows us to retain the problem's complexity while enabling us to analyze the complexity by considering only a single problem instance.

Next, we demonstrate that for any POVM base M , there is a set of unitary matrices U ( M ) satisfying that (1) P U ∼ Haar ( U ∈ U ( M )) = Ω(1) ; and (2) when the quantum states rotated by these unitary matrices, they are difficult to distinguish by the POVM base M .

At last, we only consider all the possible POVM M and their corresponding set of unitary matrix U ( M ) . By analyzing the sampling distribution for specific POVM M and unitary matrix in U ( M ) , we reduce the problem into a classical problem for resolution and provide a lower bound for the purest quantum state identification.

Similar to Definition 7 in [42], we analyze the lower bound of the error probability for any algorithm solving the purest quantum state identification problem using a 2-outcome randomly incoherent POVM to evaluate the task's difficulty.

Definition 5.1 (Randomly fixed incoherent two-outcome POVM) . We say an algorithm A with a randomly fixed incoherent two-outcome POVM, if it proceeds as the following: The algorithm A samples a POVM M = { M 0 , M 1 = I d -M 0 } from a well-designed distribution of POVMs D M and performs the two-outcome single-copy POVM M on the copies of the quantum states.

5.1 Problem reduction

In this subsection, we aim to demonstrate that if there exists a set of random quantum states T ( x ) = { τ 1 ( x ) , . . . , τ K ( x ) } which is difficult to identify the purest one in T , there also exists a corresponding set of quantum states S = { ρ 1 , . . . , ρ K } where is difficult to identify the purest one in S . In this way, we only need to construct K random quantum states, which are hard to distinguish. Then, we can demonstrate the difficulty of the purest quantum state identification problem.

To enhance our discussion, we define the problem of the purest random quantum state identification(PRQSI) with incoherent measurement as follows:

Problem 5.2 (Purest random quantum state identification(PRQSI) with incoherent measurement) . Consider a set of K unknown random quantum states, denote as T ( U ) = { τ 1 ( U ) , . . . , τ K ( U ) } , where U samples from a fixed distribution D . For each k ∈ [ K ] and U ∼ D , Tr(( τ k ( U )) 2 ) = z k . In each round t ∈ { 1 , ..., N } , the learner selects an index k t ∈ [ K ] and a POVM M t . The learner obtains a copy of τ k t ( U ) and uses M t to measure it. Upon completing N measurements, the learner selects an index k ′ ∈ [ K ] as the output. The objective of the learner is to maximize P ( k ′ ∈ arg max k ∈ [ K ] z k ) .

As shown in the following lemma, we can reduce the proof of the error probability lower bound for the PQSI problem into the proof of the lower bound for a specific instance of the PRQSI problem.

Lemma 5.3. If there exists a set of random quantum states T ( U ) in Problem 5.2 such that any algorithm A T addressing Problem 5.2 cannot identify the purest random quantum state with an error probability lower than e N , then for any algorithm A addressing Problem 1.1, there exists a specific set of quantum states S = { ρ 1 , . . . , ρ K } such that the error probability of algorithm A is not lower than e N .

Proof Sketch. Suppose that there exists an algorithm A satisfying whose error probability for solving Problem 1.1 is less than e N . We can prove that the algorithm A can solve the Problem 5.2 with

the error probability less than e N . Furthermore, we can establish the proof by considering the contrapositive of this statement. Detailed explanations of the proof are included in Appendix C.1.

According to Lemma 5.3, we will establish the lower bound of the error probability for the problem PQSI by demonstrating the error probability lower bound for the following problem:

Problem 5.4. Consider the Problem 5.2. For k ∈ [ K ] , let α k = √ dz k -1 d -1 , and τ k ( U ) = α k U | 0 ⟩⟨ 0 | U † + 1 -α k d I d , where U ∼ Haar .

Then, In the Problem 5.4, for k ∈ { 1 , ..., K } and U ∼ Haar , the purity of the quantum state τ k ( U ) satisfies:

5.2 Random quantum state purity certification

To analyze Problem 5.4, we first study the properties of the measurement results obtained from conducting N ′ measurements on the sampled quantum states from the quantum state distribution D = { ρ | ρ = αU | 0 ⟩⟨ 0 | U † + 1 -α d I d } , using a specific POVM M = { M 0 , M 1 } , where U ∼ Haar and α is a constant satisfying 0 ≤ α ≤ 1 .

Lemma 5.5. Let a ∈ [0 , 1] . Using a specific POVM M = { M 0 , M 1 } to measure the random quantum state in D = { ρ | ρ = αU | 0 ⟩⟨ 0 | U † + 1 -α d I d } . Let M = arg min M ′ ∈{ M 0 ,M 1 } Tr( M ′ ) . We have

and there is a function c ( M , U ) satisfying p M ( M | U ) -Tr( M ) d = c ( M , U ) α .

Proof Sketch. By utilizing the properties of the Haar unitary matrix, we can calculate the variance of p M ( M | U ) and prove the probabilistic bounds in the lemma using Chebyshev's inequality, thus completing the proof. The proof details are provided in Appendix C.2.

According to Lemma 5.5, for a specific POVM M and unitary matrix U , let U M denote the set of unitary matrix satisfying that U M = { U : ∣ ∣ ∣ p M ( M | U ) -Tr( M ) d ∣ ∣ ∣ < 2 α √ Tr( M ) d } . We have

P U ∼ Haar ( U ∈ U M ) ≥ 3 4 . The following analysis will focus on the unitary matrices in U M .

5.3 Error probability lower bound

In this subsection, we will prove the lower bound of error probability for using algorithms to solve Problem 1.1 and Problem 5.4.

Let M = arg min M ′ ∈{ M 0 ,M 1 } Tr( M ′ ) , then we have Tr( M ) ∈ [0 , d/ 2] . In the following theorem, let Tr( M ) > 16 in order to make Tr( M ) -2 √ Tr( M ) ≥ 1 2 Tr( M ) .

Theorem 5.6. Let M = arg min M ′ ∈{ M 0 ,M 1 } Tr( M ′ ) and Tr( M ) > 16 . For any algorithm A to solve the purest quantum state identification using fixed 2-outcome randomly incoherent POVM, there exists a set of quantum states which makes the error probability of A satisfies

where H 1 = min i ∈{ 2 ,...,K } ∆ ( i ) i .

Proof Sketch. Let p A e ( M , U ) denote the error probability for algorithm A to solve the problem 5.4, with specific unitary matrix U and POVM M . The error probability of A to solve the problem 5.4 satisfying e A N ≥ ∫ M∈D M ∫ U ∼ Haar p A e ( M , U ) 1 { U ∈ U M } d M dU .

When the i -th quantum state is measured using M , the measurement result follows a Bernoulli distribution with parameter Tr( Mρ U ) . From Lemma 5.5 and the definition of c ( M , U ) , we have Tr( Mρ i | U ) = c ( M , U ) α i + Tr( M ) .

If c ( M , U ) > 0 , we need to find the Bernoulli distribution with the largest parameter where the parameter of the i -th Bernoulli distribution is Tr( Mρ i | U ) , and we have Tr( Mρ i | U ) -Tr( Mρ j | U ) =

Since Tr( M ) > 16 and according to the definition of U ( M ) and M , for U ∈ U we have Tr( Mρ i | U ) ∈ [ Tr( M ) 2 d , 1 -Tr( M ) 2 d ] , and 1 -Tr( M ) 2 d ≥ 1 2 . Then, according to the problem of best arm identification problem with Bernoulli distribution, we can demonstrate that p A e ( U, M ) ≥ exp ( -O ( NH 1 d )) . Then we have e A N ≥ exp ( -O ( NH 1 d )) .

For any algorithm A D addressing Problem 5.4 cannot identify the purest random quantum state with an error probability lower than exp ( -O ( NH 1 d )) . According to Lemma 5.3, we can complete the proof. The proof details are provided in Appendix C.3.

6 PQSI with coherent measurement

In this section, we investigate the problem of purest quantum state identification with coherent measurement and propose an algorithm to solve the purest quantum state identification with coherent measurement based on the SWAP test.

The SWAP test is a quantum algorithm designed to assess the similarity between two quantum states. It offers a method for estimating these states' fidelity to quantify their closeness. We use the SWAP test in Figure 1 to estimate the purity of the quantum state ρ in the unknown quantum state set S . The measurement results in Figure 1 have a probability of 1+Tr( ρ 2 ) 2 for 0 and a probability of 1 -Tr( ρ 2 ) 2 for 1. The details of the algorithm are shown in Algorithm 2.

ρ

SWAP

ρ

|

0

⟩

H

•

H

Algorithm 2 Coherent measurement based algorithm for solving PQSI problem (CM-PQSI)

Input: Copy access to S = { ρ 1 , . . . , ρ K } , sample number N . Initialization: Set S 0 = { ρ 1 , . . . , ρ K } , log( K ) = 1 2 + ∑ K i =2 1 i , N 0 = 0 and N k = ⌈ 1 2log( K ) N -K K +1 -k ⌉ , for k ∈ { 1 , . . . , K -1 } .

for

i

= 1

, . . . , K

-

1

do

For all σ ∈ S i -1 , use SWAP test as Figure 1 for N k -N k -1 rounds, and set the outputs as x ( σ,N k -1 +1) , . . . , x ( σ,N k ) .

end for

Output the quantum state ρ ′ in S k -1 .

Theorem 6.1. The probability of error of Algorithm 2 satisfies

Proof Sketch. The outputs of the SWAP test are within the range [0 , 1] and are independent. Thus, we can apply the Hoeffding inequality to complete the proof. Detailed explanations of the proof are included in Appendix D.

To further clarify the role of coherent measurements, we compare the theoretical performance of the proposed algorithms both with and without the SWAP test. The essential difference lies in the scaling behavior of the error probability bounds across different purity-gap regimes.

When the purity gaps satisfy ∆ ( i ) > 1 /d 2 , the lower bound for the algorithm without the SWAP test is

In contrast, the algorithm with the SWAP test (i.e., using coherent measurements) achieves

which eliminates the additional 2 n factor in the denominator. This implies that the coherentmeasurement-based algorithm can attain the same error probability with only O (1 / 2 n ) of the sample complexity, thereby demonstrating a substantial advantage in this regime.

When the purity gaps are very small ( ∆ ( i ) ≤ 1 /d 2 ), both algorithms yield similar lower bounds of the form

and thus exhibit comparable asymptotic performance. Nevertheless, even in this regime, the algorithm employing the SWAP test tends to achieve a smaller constant factor in the exponent, resulting in empirically faster convergence and lower error rates for finite sample sizes. Therefore, the coherent measurement strategy remains effective in practice, providing improvements in both asymptotic and finite-sample efficiency.

In summary, the advantage of the SWAP test becomes particularly evident when the purity differences between quantum states are not vanishingly small. In most realistic scenarios, such tiny gaps can be regarded as negligible, and the corresponding states can be considered effectively equivalent. Consequently, for practical tasks that aim to distinguish states with noticeable purity differences, the coherent measurement strategy via the SWAP test offers a clear improvement in sample efficiency. This observation also highlights the essential role of quantum memory in enabling coherent measurements.

7 Conclusion and Outlook

In this study, we propose a pivotal problem in quantum testing, termed purest quantum state identification (PQSI). This framework applies to various quantum computing and quantum communication tasks. We develop two distinct algorithms to address this problem under different settings. When the learner utilizes incoherent (single-copy) measurement, the upper bound on the error probability of our algorithm is given by exp ( -Ω ( NH 1 log( K )2 n )) . When the learner is allowed to use coherent

(two-copy) measurement, the upper bound on the error probability is given by exp ( -Ω ( NH 2 log( K ) )) . By examining the error probabilities of these two algorithms, we can discern the advantage of the coherent measurement over the incoherent one. Furthermore, we establish that for any algorithm utilizing a randomly fixed incoherent two-outcome POVM to solve the PQSI, its error probability is lower bounded by exp ( -O ( NH 1 2 n )) . Our results lay the groundwork for further investigations into the best quantum state identification. We aim to establish a lower bound for PQSI problems across all POVM bases in future work. Several open questions remain to be addressed, including identifying the nearest quantum state with minimal trace distance and achieving the best quantum state identification with fixed confidence.

References

Peter W Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Proceedings 35th annual symposium on foundations of computer science , pages 124-134. Ieee, 1994.

Andrew J Daley, Immanuel Bloch, Christian Kokail, Stuart Flannigan, Natalie Pearson, Matthias Troyer, and Peter Zoller. Practical quantum advantage in quantum simulation. Nature , 607(7920):667-676, 2022.

Jonathan Wei Zhong Lau, Kian Hwee Lim, Harshank Shrotriya, and Leong Chuan Kwek. Nisq computing: where are we and where do we go? AAPPS bulletin , 32(1):27, 2022.

John Preskill. Quantum computing in the nisq era and beyond. Quantum , 2:79, 2018.

google. Suppressing quantum errors by scaling a surface code logical qubit. Nature , 614(7949):676-681, 2023.

Laszlo Gyongyosi and Sandor Imre. A survey on quantum computing technology. Computer Science Review , 31:51-71, 2019.

Kean Chen, Qisheng Wang, Peixun Long, and Mingsheng Ying. Unitarity estimation for quantum channels. IEEE Transactions on Information Theory , 69(8):5116-5134, 2023.

M. S. Anwar, D. Blazina, H. A. Carteret, S. B. Duckett, T. K. Halstead, J. A. Jones, C. M. Kozak, and R. J. K. Taylor. Preparing high purity initial states for nuclear magnetic resonance quantum computing. Phys. Rev. Lett. , 93:040501, Jul 2004.

Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al. Variational quantum algorithms. Nature Reviews Physics , 3(9):625-644, 2021.

Nengkun Yu. Sample efficient identity testing and independence testing of quantum states. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021) . Schloss DagstuhlLeibniz-Zentrum für Informatik, 2021.

Khabat Heshami, Duncan G England, Peter C Humphreys, Philip J Bustard, Victor M Acosta, Joshua Nunn, and Benjamin J Sussman. Quantum memories: emerging applications and recent advances. Journal of modern optics , 63(20):2005-2028, 2016.

Fermi Ma and Hsin-Yuan Huang. How to construct random unitaries. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing , pages 806-809, 2025.

Laura Cui, Thomas Schuster, Fernando Brandao, and Hsin-Yuan Huang. Unitary designs in nearly optimal depth. arXiv preprint arXiv:2507.06216 , 2025.

Thomas Schuster, Jonas Haferkamp, and Hsin-Yuan Huang. Random unitaries in extremely low depth. Science , 389(6755):92-96, 2025.

Jeongwan Haah, Yunchao Liu, and Xinyu Tan. Efficient approximate unitary designs from random pauli rotations. In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) , pages 463-475. IEEE, 2024.

Chi-Fang Chen, Jordan Docter, Michelle Xu, Adam Bouland, Fernando GSL Brandão, and Patrick Hayden. Efficient unitary designs from random sums and permutations. In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) , pages 476-484. IEEE, 2024.

Ashley Montanaro and Ronald de Wolf. A survey of quantum property testing. arXiv preprint arXiv:1310.2035 , 2013.

Dorit Aharonov, Jordan Cotler, and Xiao-Liang Qi. Quantum algorithmic measurement. Nature communications , 13(1):887, 2022.

Konrad Banaszek, Marcus Cramer, and David Gross. Focus on quantum tomography. New Journal of Physics , 15(12):125020, 2013.

David Gross, Yi-Kai Liu, Steven T Flammia, Stephen Becker, and Jens Eisert. Quantum state tomography via compressed sensing. Physical review letters , 105(15):150401, 2010.

Sitan Chen, Brice Huang, Jerry Li, Allen Liu, and Mark Sellke. When does adaptivity help for quantum state learning? In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) , pages 391-404. IEEE, 2023.

Sitan Chen, Jordan Cotler, Hsin-Yuan Huang, and Jerry Li. Exponential separations between learning with and without quantum memory. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS) , pages 574-585. IEEE, 2022.

Sebastien Bubeck, Sitan Chen, and Jerry Li. Entanglement is necessary for optimal quantum property testing. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS) , pages 692-703. IEEE, 2020.

Sitan Chen, Jerry Li, Brice Huang, and Allen Liu. Tight bounds for quantum state certification with incoherent measurements. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) , pages 1205-1213. IEEE, 2022.

John Wright. How to learn a quantum state . PhD thesis, Carnegie Mellon University, 2016.

Sitan Chen, Jerry Li, and Ryan O'Donnell. Toward instance-optimal state certification with incoherent measurements. In Conference on Learning Theory , pages 2541-2596. PMLR, 2022.

Anurag Anshu, Zeph Landau, and Yunchao Liu. Distributed quantum inner product estimation. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing , pages 44-51, 2022.

Marcel Hinsche, Marios Ioannou, Sofiene Jerbi, Lorenzo Leone, Jens Eisert, and Jose Carrasco. Efficient distributed inner product estimation via pauli sampling. arXiv preprint arXiv:2405.06544 , 2024.

Daiwei Zhu, Ze-Pei Cian, Crystal Noel, Andrew Risinger, Debopriyo Biswas, Laird Egan, Yingyue Zhu, Alaina M Green, C Huerta Alderete, Nhung H Nguyen, et al. Cross-platform comparison of arbitrary quantum states. Nature communications , 13(1):6620, 2022.

Hsin-Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements. Nature Physics , 16(10):1050-1057, 2020.

Antonio Anna Mele. Introduction to haar measure tools in quantum information: A beginner's tutorial. Quantum , 8:1340, 2024.

Jean-Yves Audibert and Sébastien Bubeck. Best arm identification in multi-armed bandits. In COLT-23th Conference on learning theory-2010 , pages 13-p, 2010.

Aurélien Garivier and Emilie Kaufmann. Optimal best arm identification with fixed confidence. In Conference on Learning Theory , pages 998-1027. PMLR, 2016.

Daniel Russo. Simple bayesian algorithms for best arm identification. In Conference on Learning Theory , pages 1417-1418. PMLR, 2016.

Kevin Jamieson and Robert Nowak. Best-arm identification algorithms for multi-armed bandits in the fixed confidence setting. In 2014 48th Annual Conference on Information Sciences and Systems (CISS) , pages 1-6, 2014.

Victor Gabillon, Mohammad Ghavamzadeh, and Alessandro Lazaric. Best arm identification: A unified approach to fixed budget and fixed confidence. Advances in Neural Information Processing Systems , 25, 2012.

Robert E Bechhofer. Single-stage procedures for ranking multiply-classified variances of normal populations. Technometrics , 10(4):693-714, 1968.

Edward Paulson. A sequential procedure for selecting the population with the largest mean from k normal populations. The Annals of Mathematical Statistics , pages 174-180, 1964.

Ilias Diakonikolas, Themis Gouleakis, John Peebles, and Eric Price. Collision-based testers are optimal for uniformity and closeness. Chic. J. Theor. Comput. Sci , 25:1-21, 2019.

Clément L Canonne. A survey on distribution testing: Your data is big. but is it blue? Theory of Computing , pages 1-100, 2020.

Fernando GSL Brandão, Wissam Chemissany, Nicholas Hunter-Jones, Richard Kueng, and John Preskill. Models of quantum complexity growth. PRX Quantum , 2(3):030316, 2021.

Weiyuan Gong, Jonas Haferkamp, Qi Ye, and Zhihan Zhang. On the sample complexity of purity and inner product estimation. arXiv preprint arXiv:2410.12712 , 2024.

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: The main claims made in the abstract and introduction include the contribution of the paper and are relevant to the main content.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.

It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.

2. Limitations

Question: Does the paper discuss the limitations of the work performed by the authors?

Answer: [Yes]

Justification: We mentioned the limitations of lower bound proof in the Conclusion and Outlook section.

Guidelines:

The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.

The authors are encouraged to create a separate "Limitations" section in their paper.

The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.

The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.

The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.

The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.

If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.

While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.

3. Theory assumptions and proofs

Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?

Answer: [Yes]

Justification: For each theoretical result, the paper provides the full set of assumptions and a complete (and correct) proof.

Guidelines:

The answer NA means that the paper does not include theoretical results.

All the theorems, formulas, and proofs in the paper should be numbered and crossreferenced.

All assumptions should be clearly stated or referenced in the statement of any theorems.

The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.

Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.

Theorems and Lemmas that the proof relies upon should be properly referenced.

4. Experimental result reproducibility

Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?

Answer: [NA]

Justification: The paper does not include experiments. We believe that conducting simulations will align with our theoretical results and we will consider to validate our algorithms experimentally by real quantum devices in the future.

Guidelines:

The answer NA means that the paper does not include experiments.

If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.

If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.

Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may

be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.

While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example

(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.

(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.

(c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).

(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.

5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [NA]

Justification: The paper does not include experiments requiring code.

Guidelines:

The answer NA means that paper does not include experiments requiring code.

Please see the NeurIPS code and data submission guidelines ( https://nips.cc/ public/guides/CodeSubmissionPolicy ) for more details.

While we encourage the release of code and data, we understand that this might not be possible, so 'No' is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).

The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines ( https: //nips.cc/public/guides/CodeSubmissionPolicy ) for more details.

The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.

The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why.

At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).

Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental setting/details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [NA]

Justification: The paper does not include experiments.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment statistical significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [NA]

Justification: The paper does not include experiments.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments compute resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [NA]

Justification: The paper does not include experiments.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code of ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: The research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [NA]

Justification: This paper discusses theoretical work related to the field of quantum information. There is no societal impact of the work performed

Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: This paper discusses theoretical work related to the field of quantum information. The paper poses no such risks.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [NA]

Justification: The paper does not use existing assets.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA]

Justification: The paper does not release new assets.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

The paper should discuss whether and how consent was obtained from people whose asset is used.

At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and research with human subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: The paper does not involve crowdsourcing nor research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.

15. Institutional review board (IRB) approvals or equivalent for research with human subjects

Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: The paper does not involve crowdsourcing nor research with human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

16. Declaration of LLM usage

Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.

Answer: [NA]

Justification: The core method development in this research does not involve LLMs as any important, original, or non-standard components.

Guidelines:

The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.

Please refer to our LLM policy ( https://neurips.cc/Conferences/2025/LLM ) for what should or should not be described.

A Auxiliary tools

A.1 Description of commonly used-notations

A.2 Probability inequalities for sums of bounded random variables

In this paper, we utilize the following inequalities, which are provided for the sake of completeness.

Theorem A.1 (Chebyshev's Inequality) . Let X be any random variable with expected value µ = E [ X ] and finite variance Var( X ) . Then, for any real number ε > 0 :

Theorem A.2 (Hoeffding's Inequality) . If X 1 , X 2 , ..., X n are independent with P ( a ≤ X i ≤ b ) = 1 and common mean µ then for any ε > 0

Theorem A.3 (Bernstein's Inequality) . If X 1 , ..., X n are independent bounded random variables such that E [ X i ] = 0 for all i ∈ { 1 , ..., n } and P ( | X i | ≤ c ) = 1 then, for any ϵ > 0 ,

where σ 2 = 1 n ∑ n i =1 Var( X i ) .

A.3 Properties of Haar unitary matrix

For a locally compact topological group, its Haar measure is a unique nonzero left-invariant measure (or right-invariant, depending on the formulation) under group operations. The Haar unitary matrix is the Haar measure on the unitary matrix group and is the concept of drawing unitary matrices uniformly at random. The formal definition of Haar unitary matrix is as follows:

Definition A.4. The Haar unitary matrix is the unique probability measure µ H that is both left and right invariant over the unitary matrix group, i.e., for all integrable functions f and for all unitary matrix V , we have:

For any unit column vector x ∈ C d , we have

We will use the following lemma to complete our proofs in this paper.

Lemma A.5 (see Lemma 22 of Ref.[27]) . Let A,B,C be Hermitian matrices. Then

and and

and

where P d ( π ) = ∑ d -1 i 1 ,...,i k =0 | i π -1 (1) , ..., i π -1 (4) ⟩⟨ i 1 , ..., i k | and π is permutation on 4 items.

B Proof of IMPQSI with incoherent measurement

B.1 Proof of property for purity collision

Lemma B.1. The expectation and variance of the purity estimation satisfying

Proof. The expectation of ˜ g satisfying

and

̸

̸

The expectation of ˜ g 2 satisfying

̸

Then, we have

B.2 proof of Theorem 4.1

By using the techniques similar to [27], we can prove the following lemma:

Lemma B.2 (See Lemma 16 of [27]) . The expectation of w ( ρ, k ) and ˜ g ρ,j in Algorithm 1 satisfies

and the variance of ˜ g ( ρ, j ) satisfies

By the definition of w ( · , · ) and the definition of ∆ ( · ) , we have

Since w ( · , · ) ∈ [0 , 1] , by Lemma B.2 and Bernstein's inequality, we have

By a union bound of error probability, we have

̸

By definition of N k , we have

Combining equation (28) and (29), we have

where H 1 = min i ∈{ 1 ,...,K } ∆ ( i ) i .

B.3 proof of Lemma 4.2

Proof. By the definition of w ( · , · ) and the definition of ∆ ( · ) , we have

Since w ( · , · ) ∈ [0 , 1] , by Lemma B.2 and Bernstein's inequality, we have

By a union bound of error probability, we have

By definition of N k , and combining equation (31), we have

where H 1 = min i ∈{ 2 ,...,K } ∆ ( i ) , and H 2 = min i ∈{ 2 ,...,K } ∆ 2 ( i ) .

C Proof of lower bound

C.1 Proof of Lemma 5.3

Suppose that there exists such an algorithm A satisfying that the error probability of A for solving Problem 1.1 with the quantum state set S ρ whose error probability is less than e N . Let p S ρ ( x 1 , y 1 ; ... ; x N , y N ; z ) denote the probability of the event satisfying

for i ∈ 1 , ..., N , in the round N , the algorithm A select the x i -th quantum state for measurement, and its output is y i ;

the algorithm A output z -th quantum state at the end.

Furthermore, let q S ρ ( x 1 , y 1 ; ... ; x N , y N ; z ) denote the error probability corresponding to p S ρ ( x 1 , y 1 ; ... ; x N , y N ; z ) .

Since when using A to solve the Problem 1.1, error probability is less than e N . Then for all quantum state set S = { ρ 1 , ..., ρ K } , we have

When the learner use the algorithm A to solve the problem 5.2, its error probability satifying

(33)

then we can prove that if there is an algorithm A can solve the Problem 5.2 with the error probability less than e N , then it can solve the Problem 1.1 with the error probability less than e N . Furthermore, we can establish the proof by considering the contrapositive of this statement.

C.2 proof of Lemma 5.5

Without loss of generality, assume that M 0 = arg min M ′ ∈{ M 0 ,M 1 } Tr( M ′ ) . According to the definition of POVM, there exists a unitary matrix V and a diagonal matrix Σ 0 = diag( b 0 , ..., b d -1 ) , where b 0 , ..., b d -1 ∈ [0 , 1] such that

We have

Let p M ( M | U ) denote the probability that M 'accepts" the quantum state αU | 0 ⟩⟨ 0 | U † + 1 -α d -1 I d . According to the property of the Haar measure and the identity matrix I d , we have

Let V i is the matrix satisfying that V i | i ⟩ = | 0 ⟩ , then V i is an unitary matrix and V -1 i = V † i , we have

̸

̸

According to the Lemma A.5, we have for i, j ∈ { 0 , ..., d -1 } , i = j ,

and and similarly

According to Equation (36), (37),(38) and (39), we have

̸

and

Then the variance of p M ( M 0 | U ) is given by

From Chebyshev's Inequality, we have

And

Let c ( M , U ) = Tr ( ∑ d -1 i =0 b i V | i ⟩⟨ i | V † ( U | 0 ⟩⟨ 0 | U † + 1 d I d ) ) , we have

C.3 Proof of Theorem 5.6

We will use the following theorem to complete the proof:

Theorem C.1 (see Theorem 4 of Ref. [32]) . Let ν 1 , ..., ν K be Bernoulli distributions with parameters in [ a, 1 -a ] , a ∈ (0 , 1 / 2) . For any forecaster, there exists a permutation σ : { 1 , ..., K } → { 1 , ..., K } such that the probability error of the forecaster on the bandit problem defined by ˜ ν 1 = ν σ (1) , ..., ˜ ν K = ν σ ( K ) satisfies

where H = min i ( E [ ν ∗ ] -E [ ν ( i ) ]) 2 i .

Let p A e ( M , U ) denote the error probability for algorithm A to solve the problem 5.4, with specific unitary matrix U and POVM M , and M = min M ′ ∈{ M 0 ,M 1 } Tr( M ′ ) . Then the error probability of A to solve the problem 5.4 satisfying

The first line corresponds to the deifintion of e A N , the second line corresponds to that p A e ( M , U ) ≥ 0 and U M is a subset of unitary matrix.

When the i -th quantum state is measured using M , the process in which the output result is accepted by M follows a Bernoulli distribution with parameter Tr( Mρ U ) . From Lemma 5.5 and the definition of c ( M , U ) , we have

If c ( M , U ) > 0 , we need to find the Bernoulli distribution with the largest parameter where the parameter of the i -th Bernoulli distribution is Tr( Mρ i | U ) = c ( M , U ) α i + Tr( M ) d . Then we have

Since Tr( M ) > 16 , for U ∈ U ( M ) we have

And since M = arg min M ′ ∈{ M 0 ,M -1 } Tr( M ′ ) , we have Tr( M ) ≤ 1 2 , then for U ∈ U we have

According to Theorem C.1 and the definition of U M , for U ∈ U M we have

According to the definition of U M , for U ∈ U M we have

According to Equation (53) and Equation (54), we have

Similarly, if c ( U, M ) < = 0 , we have

and

According to Equation (47), (55), (56) we have

Then for any algorithm A D addressing Problem 5.4 cannot identify the purest random quantum state with an error probability lower than exp ( -O ( NH 1 d )) . According to Lemma 5.3, we can complete the proof.

D Proof of Theorem 6.1

By the definition of w ( · , · ) and the definition of ∆ ( · ) , we have

Since x ( · , · ) ∈ [0 , 1] and Hoeffding's inequality, we have

By a union bound of error probability, we have

By definition of N k , we have

Combining equation (58) and (59), we have

E Simulation Results

To empirically validate our theoretical results, we performed numerical simulations on classical simulators to evaluate the performance of the proposed method under finite-sample conditions. We considered a collection of 10 randomly generated 6-qubit quantum states of the form

where each | ψ i ⟩ is a pure state sampled uniformly at random, and the mixing parameter λ i is chosen such that the purity Tr( ρ 2 i ) = 0 . 5 + 0 . 04 i . The task is to identify the state with the highest purity given N = 30 , 000 copies of the unknown quantum states.

We tested the following three algorithms:

IM_PQSI: our proposed adaptive algorithm using only incoherent measurements.

CM_PQSI: a variant incorporating coherent measurements via the SWAP test.

Unadaptive: a non-adaptive baseline that uniformly allocates samples and uses the same purity estimator as IM_PQSI.

Each algorithm was run for 100 independent trials. The results are summarized as follows:

IM_PQSI achieves a 53% success rate, with lower average purity 0.8736 and higher variance 0.001319.

CM_PQSI achieves perfect success in all 100 runs and consistently selects the state with purity 0.9.

Unadaptive achieves a 43% success rate, with lower average purity 0.86192 and higher variance 0.002145.

These results demonstrate that coherent measurements (via the SWAP test) substantially improve accuracy, confirming the theoretical prediction of their advantage. Moreover, comparing IM_PQSI with the Unadaptive algorithm shows that adaptivity enhances performance by allocating more samples to promising candidates, leading to higher estimation accuracy and success rates.

Due to the computational cost of high-dimensional quantum-state simulation, our experiments are restricted to small systems ( n = 6 ) and moderate sample budgets ( N = 3 × 10 4 ). Nevertheless, the observed behavior aligns well with our theoretical predictions, and we expect the advantages of adaptive and coherent measurement strategies to become even more pronounced for larger-scale systems and higher sample budgets.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the problem of Purest Quantum State Identification (PQSI). Given $K$ unknown $n$-qubit states, the goal is to identify the quantum state with the higest purity using $N$ total copies of these states. The authors propose algorithms for both incoherent (single-copy) and coherent (multi-copy) measurement strategies and provide upper bounds on error probability. They also establish a lower bound for the incoherent case.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The formulation of the PQSI problem is novel and opens a promising direction for integrating quantum computing with online learning. While both components (purity estimation in quantum computing and best-arm identification in online learning) are well studied individually, their combination is still non-trivial and requires careful algorithmic and theoretical design.\n2. The paper provides rigorous theoretical results, including upper bounds for both incoherent and coherent measurement strategies, and a lower bound using carefully constructed hardness reductions.\n3. Compared with existing studies on quantum online learning, this paper adopts a more natural problem formulation that does not rely on strong quantum oracles.\n\nWeaknesses:\n1. The paper lacks empirical or simulation-based validation to support the theoretical claims.\n2. The presentation lacks key intuitions. For example, the collision estimator is conceptually simple, but its physical meaning and intuitions are not explained when presenting the algorithms.\n\nOverall, I recommend accepting this paper. Integrating quantum computing and online learning is a timely and important topic, but existing studies usually rely on strong assumptions about quantum oracles, and the feedback models lack concrete counterparts in real-world quantum systems. In contrast, this paper proposes a more practical problem formulation.\n\nQUESTIONS:\n1. Have you considered the fixed-confidence setting? Could your algorithms be adapted to this setting rather than the fixed-budget framework?\n2. Have you considered the lower bound for coherent measurements? How does it differ from the incoherent case, and what challenges arise in establishing it?\n3. Is it possible to use alternative purity estimation methods such as quantum tomography? If so, how would this affect the performance of your algorithms?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose two algorithms for purest quantum state identification. The first algorithm requires first a random unitary transformation of the quantum state, followed by a POVM measurement. Purity can be estimated by the probability of identical outcomes, after which the highest is found. The second algorithm utilizes the SWAP test to directly compute the purity instead of estimation.\n\nSTRENGTHS AND WEAKNESSES:\n__Strengths__\n1. The paper proposes an interesting question for quantum information.\n\n2. The paper provides a lower bound in addition to algorithm results, adding more context to the performance of the algorithm itself.\n\n__Weakness__\n1. Not sure if the scope of the paper is aligned with the call for papers, particularly the theory track, though I suppose optimization could work.\n\n2. Algorithm and results seem simililar to the classical shadows method proposed in Ref. [26]. See further Question 1.\n\n3. The proof of requiring unitary 4-designs is not provided in the manuscript, and Haar random unitaries are assumed to be provided in the proof. See further Question 2.\n\nQUESTIONS:\n1. The algorithm for incoherent measurements seems to share similiarities with the classical shadows method proposed in Ref. [26] with slight modifications in for 4-unitary designs, projective measurements to POVM, and from estimating all values to an fixed accuracy to a search problem. Regardless, the main protocol seems similiar, with the random transformation and subsequent channel inversion that takes the form of computing identical probablities in this algorithm.\n\n2. Given that Lemma A.5 only lists results up to the 3rd power, does this imply that the protocol only require 3-designs instead of 4-designs? In this case, we would only require random Clifford circuits, which would be surprising.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces Purest Quantum State Identification (PQSI) - the task of identifying the purest state out of a set of finite states. The authors propose algorithms for both single-copy and multi-copy measurement strategies, along with a lower bound for the former. The motivation is to identify the quantum state least affected by noise, which is framed as identifying the state with the highest purity.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper contains algorithms for both single and multi-copy measurement ability.\n- The authors keep practical considerations in mind. \n\nWeaknesses:\n1.  Before delving into technical flaws, I believe this work is simply not suited for NeurIPS. The content and research direction has virtually nothing to do with machine learning. This type of work is probably better suited for a quantum theory journal/conference like QIP/TQC.\n\n2.  The derived bounds on the error probability are all extremely loose. Am I perhaps missing something? For example in Thm 1.3 and 1.4, if the gaps H_1 and H_2 are small, this gives an upper bound close to 1. These are two of the main results in a completely theory-based paper.\n\n3.  There are many technical issues in the proofs and I question the validity of the technical soundness.\n\nUpdate after rebuttal: \n1.  Indeed, this work is suitable for the online learning theory category, but I still maintain that it is more appropriate for QIP/TQC. \n2.  The authors clarified that loose bounds in small gap regimes are an illustration of the hardness of the task and pointed to other analogous settings in RL for which this occurs which marginally puts this concern at ease.\n3. My concerns around technical issues have been resolved by the author's response.\n\nQUESTIONS:\n- The distinction between properties for a fixed POVM and properties averaged over Haar random POVMs is sometimes mixed up by the authors. This can drastically affect the results and their proofs require careful attention to how expectations and variances are taken.\n- How do the authors get the second equation of 552? This seems incorrect. \n- In particular, there are many unjustified steps/approximations in proofs, especially in Lemma B.1 (the variance approximation) and Lemma 5.5 (factor of 2 in variance and justification for Haar integral application is questionable). I suggest the authors carefully redo their technical analyses. \n- Notation could be more consistent.\n\nUpdate: these questions and concerns have been addressed in the comments and I am satisfied with the answers.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

comments: SUMMARY:
This paper addresses the problem of quantum noise in the context of quantum system identification and introduces the task of purest quantum state identification. Theoretical analysis is provided to support the proposed approach.

STRENGTHS AND WEAKNESSES:
Strengths:
The authors introduce an incoherent measurement algorithm to solve the purest quantum state identification problem and provide a theoretical analysis of the error probability and lower bound.
Weaknesses:
No simulation experiments are presented. Including such results would help validate the effectiveness of the proposed algorithm.

QUESTIONS:
1）Regarding the lower bound, it would be helpful to clarify which specific factors or parameters determine the lower bound in the proposed analysis. For example, does it depend on the system dimension, noise level, or properties of the quantum state ensemble?
2）In Algorithm 2, an additional SWAP test circuit is introduced to perform coherent measurements. However, it remains unclear how this component contributes to improving the accuracy or robustness of purest quantum state identification. A more detailed discussion or theoretical justification would strengthen the argument.
3）Experimental validation of the proposed method is currently lacking. Incorporating numerical simulations or prototype experiments—such as using noisy intermediate-scale quantum (NISQ) devices or classical simulators—would significantly enhance the applicability of the approach.

LIMITATIONS:
Including simulation experiments would further strengthen the theoretical claims and improve the overall completeness of the work.
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03