Original: Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants
Revised: Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants

Original: Abstract
Revised: Abstract

Original: Given a convex function f : R d → R , the problem of sampling from a distribution ∝ e − f ( x ) is called log-concave sampling.
Revised: Given a convex function f : R d → R , the problem of sampling from a distribution ∝ e − f ( x ) is called log-concave sampling.

Original: This task has wide applications in machine learning, physics, statistics, etc.
Revised: This task has wide applications in machine learning, physics, statistics, etc.

Original: In this work, we develop quantum algorithms for sampling log-concave distributions and for estimating their normalizing constants (cid:82) R d e − f ( x ) d x .
Revised: In this work, we develop quantum algorithms for sampling log-concave distributions and for estimating their normalizing constants (cid:82) R d e − f ( x ) d x .

Original: First, we use underdamped Langevin diffusion to develop quantum algorithms that match the query complexity (in terms of the condition number κ and dimension d ) of analogous classical algorithms that use gradient (ﬁrst-order) queries, even though the quantum algorithms use only evaluation (zeroth-order) queries.
Revised: First, we use underdamped Langevin diffusion to develop quantum algorithms that match the query complexity (in terms of the condition number κ and dimension d ) of analogous classical algorithms that use gradient (ﬁrst-order) queries, even though the quantum algorithms use only evaluation (zeroth-order) queries.

Original: For estimating normalizing constants, these algorithms also achieve quadratic speedup in the multiplicative error (cid:15) .
Revised: For estimating normalizing constants, these algorithms also achieve quadratic speedup in the multiplicative error (cid:15) .

Original: Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity (cid:101) O
Revised: Second, we develop quantum Metropolis-adjusted Langevin algorithms with query complexity (cid:101) O

Original: ( κ 1 / 2 d ) and (cid:101) O ( κ 1 / 2 d 3 / 2 /(cid:15) ) for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in κ, d, (cid:15) over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks.
Revised: ( κ 1 / 2 d ) and (cid:101) O ( κ 1 / 2 d 3 / 2 /(cid:15) ) for log-concave sampling and normalizing constant estimation, respectively, achieving polynomial speedups in κ, d, (cid:15) over the best known classical algorithms by exploiting quantum analogs of the Monte Carlo method and quantum walks.

Original: We also prove a 1 /(cid:15) 1 − o (1) quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in (cid:15) .
Revised: We also prove a 1 /(cid:15) 1 − o (1) quantum lower bound for estimating normalizing constants, implying near-optimality of our quantum algorithms in (cid:15) .

Original:  Introduction
Revised:  Introduction

Original: Sampling from a given distribution is a fundamental computational problem.
Revised: Sampling from a given distribution is a fundamental computational problem.

Original: For example, in statistics, samples can determine conﬁdence intervals or explore posterior distributions.
Revised: For example, in statistics, samples can determine conﬁdence intervals or explore posterior distributions.

Original: In machine learning, samples are used for regression and to train supervised learning models.
Revised: In machine learning, samples are used for regression and to train supervised learning models.

Original: In optimization, samples from well-chosen distributions can produce points near local or even global optima.
Revised: In optimization, samples from well-chosen distributions can produce points near local or even global optima.

Original: Sampling can be nontrivial even when the distribution is known.
Revised: Sampling can be nontrivial even when the distribution is known.

Original: Indeed, efﬁcient sampling is often a challenging computational problem, and bottlenecks the running time in many applications.
Revised: Indeed, efﬁcient sampling is often a challenging computational problem, and bottlenecks the running time in many applications.

Original: Many efforts have been made to develop fast sampling methods.
Revised: Many efforts have been made to develop fast sampling methods.

Original: Among those, one of the most successful tools is Markov Chain Monte Carlo (MCMC), which uses a Markov chain that converges to the desired distribution to (approximately) sample from it.
Revised: Among those, one of the most successful tools is Markov Chain Monte Carlo (MCMC), which uses a Markov chain that converges to the desired distribution to (approximately) sample from it.

Original: Here we focus on the fundamental task of log-concave sampling , i.e., sampling from a distribution proportional to e − f where f : R d → R is a convex function.
Revised: Here we focus on the fundamental task of log-concave sampling , i.e., sampling from a distribution proportional to e − f where f : R d → R is a convex function.

Original: This covers many practical applications such as multivariate Gaussian distributions and exponential distributions.
Revised: This covers many practical applications such as multivariate Gaussian distributions and exponential distributions.

Original: Provable performance guarantees for log-concave sampling have been widely studied [14].
Revised: Provable performance guarantees for log-concave sampling have been widely studied [15].

Original: A closely related problem is estimating the normalizing constants of log-concave distributions, which also has many applications [15].
Revised: A closely related problem is estimating the normalizing constants of log-concave distributions, which also has many applications [16].

Original: Quantum computing has been applied to speed up many classical algorithms based on Markov processes, so it is natural to investigate quantum algorithms for log-concave sampling.
Revised: Quantum computing has been applied to speed up many classical algorithms based on Markov processes, so it is natural to investigate quantum algorithms for log-concave sampling.

Original: If we can prepare a quantum state whose amplitudes are the square roots of the corresponding probabilities, then measurement yields a random sample from the desired distribution.
Revised: If we can prepare a quantum state whose amplitudes are the square roots of the corresponding probabilities, then measurement yields a random sample from the desired distribution.

Original: In this approach, the number of required qubits is only poly-logarithmic in the size of the sample space.
Revised: In this approach, the number of required qubits is only poly-logarithmic in the size of the sample space.

Original: Unfortunately, such a quantum state probably cannot be efﬁciently prepared in general, since this would imply SZK ⊆
Revised: Unfortunately, such a quantum state probably cannot be efﬁciently prepared in general, since this would imply SZK ⊆

Original: BQP [1].
Revised: BQP [1].

Original: Nevertheless, in some cases, quantum algorithms can achieve polynomial speedup over classical algorithms.
Revised: Nevertheless, in some cases, quantum algorithms can achieve polynomial speedup over classical algorithms.

Original: Examples include uniform sampling on a 2D lattice [35], estimating partition functions [4, 21, 31, 45, 46], and estimating volumes of convex bodies [6].
Revised: Examples include uniform sampling on a 2D lattice [35], estimating partition functions [4, 22, 31, 45, 46], and estimating volumes of convex bodies [6].

Original: However, despite the importance of sampling log-concave distributions and estimating normalizing constants, we are not aware of any previous quantum speedups for general instances of these problems.
Revised: However, despite the importance of sampling log-concave distributions and estimating normalizing constants, we are not aware of any previous quantum speedups for general instances of these problems.

Original: Formulation In this paper, we consider a d -dimensional convex function f : R d → R which is
Revised: Formulation In this paper, we consider a d -dimensional convex function f : R d → R which is

Original: L -smooth and µ -strongly convex, i.e., µ, L > 0 and for any x, y ∈ R d , x (cid:54) =
Revised: L -smooth and µ -strongly convex, i.e., µ, L > 0 and for any x, y ∈ R d , x (cid:54) =

Original: y ,
Revised: y ,

Original: [Equation]
Revised: [Equation]

Original: We denote by κ :=
Revised: We denote by κ :=

Original: L/µ the condition number of f .
Revised: L/µ the condition number of f .

Original: The corresponding log-concave distribution has probability density ρ f : R d → R with
Revised: The corresponding log-concave distribution has probability density ρ f : R d → R with

Original: [Equation]
Revised: [Equation]

Original: where the normalizing constant is
Revised: where the normalizing constant is

Original: [Equation]
Revised: [Equation]

Original: When there is no ambiguity, we abbreviate ρ f and Z f as ρ and Z , respectively.
Revised: When there is no ambiguity, we abbreviate ρ f and Z f as ρ and Z , respectively.

Original: Given an (cid:15) ∈ (0 , 1) , • the goal of log-concave sampling is to output a random variable with distribution (cid:101) ρ such that
Revised: Given an (cid:15) ∈ (0 , 1) , • the goal of log-concave sampling is to output a random variable with distribution (cid:101) ρ such that

Original: (cid:107) (cid:101) ρ − ρ (cid:107) ≤ (cid:15)
Revised: (cid:107) (cid:101) ρ − ρ (cid:107) ≤ (cid:15)

Original: , and the goal of normalizing constant estimation is to output a value
Revised: , and the goal of normalizing constant estimation is to output a value

Original: (cid:101)
Revised: (cid:101)

Original: Z such that with probability at
Revised: Z such that with probability at

Original: least 2 / 3 , (1 − (cid:15) )
Revised: least 2 / 3 , (1 − (cid:15) )

Original: Z ≤ (cid:101) Z ≤ (1 + (cid:15) )
Revised: Z ≤ (cid:101) Z ≤ (1 + (cid:15) )

Original: Z
Revised: Z

Original: Here (cid:107) · (cid:107) is a certain norm.
Revised: Here (cid:107) · (cid:107) is a certain norm.

Original: We consider the general setting where the function f is speciﬁed by an oracle.
Revised: We consider the general setting where the function f is speciﬁed by an oracle.

Original: In particular, we consider the quantum evaluation oracle O f , a standard model in the quantum computing literature [3, 6, 7, 50].
Revised: In particular, we consider the quantum evaluation oracle O f , a standard model in the quantum computing literature [3, 6, 7, 50].

Original: The evaluation oracle acts as
Revised: The evaluation oracle acts as

Original: [Equation]
Revised: [Equation]

Original: d ,
Revised: d ,

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation]
Revised: [Equation]

Original: (Quantum computing notations are brieﬂy explained in Section 2.)
Revised: (Quantum computing notations are brieﬂy explained in Section 2.)

Original: We also consider the quantum gradient oracle
Revised: We also consider the quantum gradient oracle

Original: O ∇ f with
Revised: O ∇ f with

Original: [Equation]
Revised: [Equation]

Original: In other words, we allow superpositions of queries to both function evaluations and gradients.
Revised: In other words, we allow superpositions of queries to both function evaluations and gradients.

Original: The essence of quantum speedup is the ability to compute with carefully designed superpositions.
Revised: The essence of quantum speedup is the ability to compute with carefully designed superpositions.

Original: Contributions Our main results are quantum algorithms that speed up log-concave sampling and normalizing constant estimation.
Revised: Contributions Our main results are quantum algorithms that speed up log-concave sampling and normalizing constant estimation.

Original: Theorem 1.1 (Main log-concave sampling result) .
Revised: Theorem 1.1 (Main log-concave sampling result) .

Original: Let ρ denote the log-concave distribution (1.2).
Revised: Let ρ denote the log-concave distribution (1.2).

Original: There exist quantum algorithms that output a random variable distributed according to (cid:101) ρ such that
Revised: There exist quantum algorithms that output a random variable distributed according to (cid:101) ρ such that

Original: • W 2 ( (cid:101) ρ, ρ ) ≤ (cid:15) where W 2 is the Wasserstein 2-norm (2.4), using (cid:101) O ( κ 7 / 6 d 1 / 6 (cid:15) − 1 / 3 + κd 1 / 3 (cid:15) − 2 / 3 ) queries to the
Revised: • W 2 ( (cid:101) ρ, ρ ) ≤ (cid:15) where W 2 is the Wasserstein 2-norm (2.4), using (cid:101) O ( κ 7 / 6 d 1 / 6 (cid:15) − 1 / 3 + κd 1 / 3 (cid:15) − 2 / 3 ) queries to the

Original: quantum
Revised: quantum

Original: evaluation oracle (1.4); or • (cid:107) (cid:101) ρ − ρ (cid:107) TV ≤ (cid:15) where (cid:107) · (cid:107) TV is the total variation distance (2.3), using (cid:101) O (cid:0) κ 1 / 2 d (cid:1) queries to the quantum gradient oracle (1.5), or (cid:101) O (cid:0) κ 1 / 2 d 1 / 4 (cid:1) queries when the initial distribution is warm (formally deﬁned in Appendix C.2.1).
Revised: evaluation oracle (1.4); or • (cid:107) (cid:101) ρ − ρ (cid:107) TV ≤ (cid:15) where (cid:107) · (cid:107) TV is the total variation distance (2.3), using (cid:101) O (cid:0) κ 1 / 2 d (cid:1) queries to the quantum gradient oracle (1.5), or (cid:101) O (cid:0) κ 1 / 2 d 1 / 4 (cid:1) queries when the initial distribution is warm (formally deﬁned in Appendix C.2.1).

Original: In the above results, the query complexity (cid:101) O ( κ 7 / 6 d 1 / 6 (cid:15) − 1 / 3 + κd 1 / 3 (cid:15) − 2 / 3 ) is achieved by our quantum ULD-RMM algorithm.
Revised: In the above results, the query complexity (cid:101) O ( κ 7 / 6 d 1 / 6 (cid:15) − 1 / 3 + κd 1 / 3 (cid:15) − 2 / 3 ) is achieved by our quantum ULD-RMM algorithm.

Original: Although the quantum query complexity is the same as the best known classical result [37], we emphasize that our quantum algorithm uses a zeroth-order oracle while [37] uses a ﬁrst-order oracle.
Revised: Although the quantum query complexity is the same as the best known classical result [37], we emphasize that our quantum algorithm uses a zeroth-order oracle while [37] uses a ﬁrst-order oracle.

Original: The query complexity (cid:101) O (cid:0) κ 1 / 2 d (cid:1) is achieved by our quantum MALA algorithm that uses a ﬁrst-order oracle (as in classical algorithms).
Revised: The query complexity (cid:101) O (cid:0) κ 1 / 2 d (cid:1) is achieved by our quantum MALA algorithm that uses a ﬁrst-order oracle (as in classical algorithms).

Original: This is a quadratic speedup in κ compared with the best known classical algorithm [27].
Revised: This is a quadratic speedup in κ compared with the best known classical algorithm [28].

Original: With a warm start, our quantum speedup is even more signiﬁcant: we achieve quadratic speedups in κ and d as compared with the best known classical algorithm with a warm start [47].
Revised: With a warm start, our quantum speedup is even more signiﬁcant: we achieve quadratic speedups in κ and d as compared with the best known classical algorithm with a warm start [47].

Original: Theorem 1.2 (Main normalizing constant estimation result) .
Revised: Theorem 1.2 (Main normalizing constant estimation result) .

Original: There exist quantum algorithms that estimate the normalizing constant by (cid:101)
Revised: There exist quantum algorithms that estimate the normalizing constant by (cid:101)

Original: Z within multiplicative error (cid:15) with probability at least 3 / 4 , • using (cid:101) O ( κ 7 / 6 d 7 / 6 (cid:15) − 1 + κd 4 / 3 (cid:15) − 1 ) queries to the quantum evaluation oracle (1.4); or• using (cid:101) 
Revised: Z within multiplicative error (cid:15) with probability at least 3 / 4 , • using (cid:101) O ( κ 7 / 6 d 7 / 6 (cid:15) − 1 + κd 4 / 3 (cid:15) − 1 ) queries to the quantum evaluation oracle (1.4); or using (cid:101) O (

Original: O ( κ 1 / 2 d 3 / 2 (cid:15) − 1 ) queries to the quantum gradient oracle
Revised: κ 1 / 2 d 3 / 2 (cid:15) − 1 ) queries to the quantum gradient oracle

Original: Furthermore, this task has quantum query complexity at least Ω( (cid:15) − 1+ o (1) )
Revised: Furthermore, this task has quantum query complexity at least Ω( (cid:15) − 1+ o (1) )

Original: (Theorem 5.1).
Revised: (Theorem 5.1).

Original: Our query complexity (cid:101) O ( κ 7 / 6 d 7 / 6 (cid:15) − 1 + κd 4 / 3 (cid:15) − 1 ) for normalizing constant estimation achieves a quadratic speedup in precision compared with the best known classical algorithm [15].
Revised: Our query complexity (cid:101) O ( κ 7 / 6 d 7 / 6 (cid:15) − 1 + κd 4 / 3 (cid:15) − 1 ) for normalizing constant estimation achieves a quadratic speedup in precision compared with the best known classical algorithm [16].

Original: More remarkably, our quantum ULD-RMM algorithm again uses a zeroth-order oracle while the slower best known classical algorithm uses a ﬁrst-order oracle [15].
Revised: More remarkably, our quantum ULD-RMM algorithm again uses a zeroth-order oracle while the slower best known classical algorithm uses a ﬁrst-order oracle [16].

Original: Our quantum algorithm working with a ﬁrst-order oracle achieves polynomial speedups in all parameters compared with the best known classical algorithm [15].
Revised: Our quantum algorithm working with a ﬁrst-order oracle achieves polynomial speedups in all parameters compared with the best known classical algorithm [16].

Original: Moreover, the precision-dependence of our quantum algorithms is nearly optimal, which is quadratically better than the classical lower bound in 1 /(cid:15)
Revised: Moreover, the precision-dependence of our quantum algorithms is nearly optimal, which is quadratically better than the classical lower bound in 1 /(cid:15)

Original: To the best of our knowledge, these are the ﬁrst quantum algorithms with quantum speedup for the fundamental problems of log-concave sampling and estimating normalizing constants.
Revised: To the best of our knowledge, these are the ﬁrst quantum algorithms with quantum speedup for the fundamental problems of log-concave sampling and estimating normalizing constants.

Original: We explore multiple classical techniques including the underdamped Langevin diffusion (ULD) method [11– 13, 43], the randomized midpoint method for underdamped Langevin diffusion (ULD-RMM) [36, 37], and the Metropolis adjusted Langevin algorithm (MALA) [8, 10, 14, 27, 28,  47], and achieve quantum speedups.
Revised: We explore multiple classical techniques including the underdamped Langevin diffusion (ULD) method [12– 14, 43], the randomized midpoint method for underdamped Langevin diffusion (ULD-RMM) [36, 37], and the Metropolis adjusted Langevin algorithm (MALA) [8, 11, 15, 28, 29, 47], and achieve quantum speedups.

Original: Our main contributions are as follows
Revised: Our main contributions are as follows

Original: • Log-concave sampling.
Revised: • Log-concave sampling.

Original: For this problem, our quantum algorithms based on ULD and ULDRMM have the same query complexity as the best known classical algorithms, but our quantum algorithms only use a zeroth-order (evaluation) oracle, while the classical algorithms use the ﬁrstorder (gradient) oracle.
Revised: For this problem, our quantum algorithms based on ULD and ULDRMM have the same query complexity as the best known classical algorithms, but our quantum algorithms only use a zeroth-order (evaluation) oracle, while the classical algorithms use the ﬁrstorder (gradient) oracle.

Original: For MALA, this improvement on the order of oracles is nontrivial, but we can use the quantum gradient oracle in our quantum MALA algorithm to achieve a quadratic speedup in the condition number κ .
Revised: For MALA, this improvement on the order of oracles is nontrivial, but we can use the quantum gradient oracle in our quantum MALA algorithm to achieve a quadratic speedup in the condition number κ .

Original: Furthermore, given a warm-start distribution, our quantum algorithm achieves a quadratic speedup in all parameters.
Revised: Furthermore, given a warm-start distribution, our quantum algorithm achieves a quadratic speedup in all parameters.

Original: • Normalizing constant estimation.
Revised: • Normalizing constant estimation.

Original: For this problem, our quantum algorithms provide larger speedups.
Revised: For this problem, our quantum algorithms provide larger speedups.

Original: In particular, our quantum algorithms based on ULD and ULD-RMM achieve quadratic
Revised: In particular, our quantum algorithms based on ULD and ULD-RMM achieve quadratic

Original: speedup in the multiplicative precision (cid:15) (while using a zeroth-order oracle) compared with the corresponding best-known classical algorithms (using a ﬁrst-order oracle).
Revised: speedup in the multiplicative precision (cid:15) (while using a zeroth-order oracle) compared with the corresponding best-known classical algorithms (using a ﬁrst-order oracle).

Original: Our quantum algorithm based on MALA achieves polynomial speedups in all parameters.
Revised: Our quantum algorithm based on MALA achieves polynomial speedups in all parameters.

Original: Furthermore, we prove that our quantum algorithm is nearly optimal in terms of (cid:15) .
Revised: Furthermore, we prove that our quantum algorithm is nearly optimal in terms of (cid:15) .

Original: We summarize our results and compare them to previous classical algorithms in Table 1 and Table 2.
Revised: We summarize our results and compare them to previous classical algorithms in Table 1 and Table 2.

Original: See Appendix A for more detailed comparisons to related classical and quantum work
Revised: See Appendix A for more detailed comparisons to related classical and quantum work

Original: [Table] [Table]
Revised: [Table] [Table]

Original: TV
Revised: TV

Original: [Table]
Revised: [Table]

Original: Techniques In this work, we develop a systematic approach for studying the complexity of quantum walk mixing and show that for any reversible classical Markov chain, we can obtain quadratic speedup for the mixing time as long as the initial distribution is warm.
Revised: Techniques In this work, we develop a systematic approach for studying the complexity of quantum walk mixing and show that for any reversible classical Markov chain, we can obtain quadratic speedup for the mixing time as long as the initial distribution is warm.

Original: In particular, we apply the quantum walk and quantum annealing in the context of Langevin dynamics and achieve polynomial quantum speedups
Revised: In particular, we apply the quantum walk and quantum annealing in the context of Langevin dynamics and achieve polynomial quantum speedups

Original: [Table]
Revised: [Table]

Original: The technical ingredients of our quantum algorithms are highlighted below.
Revised: The technical ingredients of our quantum algorithms are highlighted below.

Original: • Quantum simulated annealing (Lemma 3.2).
Revised: • Quantum simulated annealing (Lemma 3.2).

Original: Our quantum algorithm for estimating normalizing constants combines the quantum simulated annealing framework of [45] and the quantum mean estimation algorithm of [31].
Revised: Our quantum algorithm for estimating normalizing constants combines the quantum simulated annealing framework of [45] and the quantum mean estimation algorithm of [31].

Original: For each type of Langevin dynamics (which are random walks), we build a corresponding quantum walk.
Revised: For each type of Langevin dynamics (which are random walks), we build a corresponding quantum walk.

Original: Crucially, the spectral gap of the random walk is quadratically ampliﬁed in the phase gap of the corresponding quantum walk.
Revised: Crucially, the spectral gap of the random walk is quadratically ampliﬁed in the phase gap of the corresponding quantum walk.

Original: This allows us to use a Grover-like procedure to produce the stationary distribution state given a sufﬁciently good initial state.
Revised: This allows us to use a Grover-like procedure to produce the stationary distribution state given a sufﬁciently good initial state.

Original: In the simulated annealing framework, this initial state is the stationary distribution state of the previous Markov chain.
Revised: In the simulated annealing framework, this initial state is the stationary distribution state of the previous Markov chain.

Original: • Effective spectral gap (Lemma C.7).
Revised: • Effective spectral gap (Lemma C.7).

Original: We show how to leverage a “warm” initial distribution to achieve a quantum speedup for sampling.
Revised: We show how to leverage a “warm” initial distribution to achieve a quantum speedup for sampling.

Original: Classically, a warm start leads to faster mixing even if
Revised: Classically, a warm start leads to faster mixing even if

Original: the spectral gap is small.
Revised: the spectral gap is small.

Original: Quantumly, we generalize the notion of “effective spectral gap” [6, 26, 34] to our more general sampling problem.
Revised: Quantumly, we generalize the notion of “effective spectral gap” [6, 27, 34] to our more general sampling problem.

Original: We show that with a bounded warmness parameter, quantum algorithms can achieve a quadratic speedup in the mixing time.
Revised: We show that with a bounded warmness parameter, quantum algorithms can achieve a quadratic speedup in the mixing time.

Original: By viewing the sampling problem as a simulated annealing process with only one Markov chain, we prove a quadratic speedup for quantum MALA by analyzing the effective spectral gap.
Revised: By viewing the sampling problem as a simulated annealing process with only one Markov chain, we prove a quadratic speedup for quantum MALA by analyzing the effective spectral gap.

Original: • Quantum gradient estimation (Lemma C.1).
Revised: • Quantum gradient estimation (Lemma C.1).

Original: We adapt Jordan’s quantum gradient algorithm [23] to the ULD and ULD-RMM algorithms and give rigorous proofs to bound the sampling error due to gradient estimation errors.
Revised: We adapt Jordan’s quantum gradient algorithm [24] to the ULD and ULD-RMM algorithms and give rigorous proofs to bound the sampling error due to gradient estimation errors.

Original: Open questions Our work raises several natural questions for future investigation:
Revised: Open questions Our work raises several natural questions for future investigation:

Original: • Can we achieve quantum speedup in d and κ for unadjusted Langevin algorithms such as ULD and ULD-RMM?
Revised: • Can we achieve quantum speedup in d and κ for unadjusted Langevin algorithms such as ULD and ULD-RMM?

Original: The main difﬁculty is that ULD and ULD-RMM are irreversible, while most available quantum walk techniques only apply to reversible Markov chains.
Revised: The main difﬁculty is that ULD and ULD-RMM are irreversible, while most available quantum walk techniques only apply to reversible Markov chains.

Original: New techniques might be required to resolve this question.
Revised: New techniques might be required to resolve this question.

Original: • Can we achieve further quantum speedup for estimating normalizing constants with a warm start distribution?
Revised: • Can we achieve further quantum speedup for estimating normalizing constants with a warm start distribution?

Original: This might require a more reﬁned version of quantum mean estimation.
Revised: This might require a more reﬁned version of quantum mean estimation.

Original: • Can we give quantum algorithms for estimating normalizing constants with query complexity sublinear in d ?
Revised: • Can we give quantum algorithms for estimating normalizing constants with query complexity sublinear in d ?

Original: Such a result would give a provable quantum-classical separation due to the Ω( d 1 − o (1) /(cid:15)
Revised: Such a result would give a provable quantum-classical separation due to the Ω( d 1 − o (1) /(cid:15)

Original: 2 − o (1) ) classical lower bound proved in [15].
Revised: 2 − o (1) ) classical lower bound proved in [16].

Original: Limitations and
Revised: Limitations and

Original: societal impacts Researchers working on theoretical aspects of quantum computing or Monte Carlo methods may beneﬁt from our results.
Revised: societal impacts Researchers working on theoretical aspects of quantum computing or Monte Carlo methods may beneﬁt from our results.

Original: In the long term, once fault-tolerant quantum computers have been built, our results may ﬁnd practical applications in MCMC methods arising in the real world.
Revised: In the long term, once fault-tolerant quantum computers have been built, our results may ﬁnd practical applications in MCMC methods arising in the real world.

Original: As far as we are aware, our work does not have negative societal impacts
Revised: As far as we are aware, our work does not have negative societal impacts

Original:  Preliminaries
Revised:  Preliminaries

Original: Basic deﬁnitions of quantum computation Quantum mechanics is formulated in terms of linear algebra.
Revised: Basic deﬁnitions of quantum computation Quantum mechanics is formulated in terms of linear algebra.

Original: The computational basis of C d is { (cid:126)e 0 , . . .
Revised: The computational basis of C d is { (cid:126)e 0 , . . .

Original: ,(cid:126)e d − 1 } , where (cid:126)e i = (0 , . . .
Revised: ,(cid:126)e d − 1 } , where (cid:126)e i = (0 , . . .

Original: . , 0) (cid:62) with the 1 in the ( i + 1) st position.
Revised: . , 0) (cid:62) with the 1 in the ( i + 1) st position.

Original: We use Dirac notation , writing | i (cid:105) (called a “ket”) for (cid:126)e i and (cid:104) i | (a “bra”) for (cid:126)e (cid:62) i .
Revised: We use Dirac notation , writing | i (cid:105) (called a “ket”) for (cid:126)e i and (cid:104) i | (a “bra”) for (cid:126)e (cid:62) i .

Original: The tensor product of quantum states is their Kronecker product:
Revised: The tensor product of quantum states is their Kronecker product:

Original: if | u (cid:105) ∈
Revised: if | u (cid:105) ∈

Original: C
Revised: C

Original: d 1 and | v (cid:105) ∈ C d 2 , then we have | u (cid:105) ⊗ | v (cid:105) ∈
Revised: d 1 and | v (cid:105) ∈ C d 2 , then we have | u (cid:105) ⊗ | v (cid:105) ∈

Original: C d 1 ⊗ C d 2 with
Revised: C d 1 ⊗ C d 2 with

Original: [Equation]
Revised: [Equation]

Original: The basic element of quantum information is a qubit , a quantum state in C 2 , which can be written as a | 0 (cid:105)
Revised: The basic element of quantum information is a qubit , a quantum state in C 2 , which can be written as a | 0 (cid:105)

Original: + b | 1 (cid:105) for some a, b ∈ C with | a | 2 + | b | 2 = 1 .
Revised: + b | 1 (cid:105) for some a, b ∈ C with | a | 2 + | b | 2 = 1 .

Original: An n -qubit tensor product state can be written as | v 1 (cid:105) ⊗ · · · ⊗ | v n (cid:105) ∈ C 2 n , where for any i ∈ [ n ] , | v i (cid:105) is a one-qubit state.
Revised: An n -qubit tensor product state can be written as | v 1 (cid:105) ⊗ · · · ⊗ | v n (cid:105) ∈ C 2 n , where for any i ∈ [ n ] , | v i (cid:105) is a one-qubit state.

Original: Note however that most states in C 2 n are not product states.
Revised: Note however that most states in C 2 n are not product states.

Original: We sometimes abbreviate | u (cid:105) ⊗ | v (cid:105) as | u (cid:105)| v (cid:105) .
Revised: We sometimes abbreviate | u (cid:105) ⊗ | v (cid:105) as | u (cid:105)| v (cid:105) .

Original: Operations on quantum states are unitary transformations .
Revised: Operations on quantum states are unitary transformations .

Original: They are typically stated in the circuit model, where a k -qubit gate is a unitary matrix in C 2 k .
Revised: They are typically stated in the circuit model, where a k -qubit gate is a unitary matrix in C 2 k .

Original: Two-qubit gates are universal , i.e., every n -qubit gate can be decomposed into a product of gates that act as the identity on n − 2 qubits and as some two-qubit gate on the other 2 qubits.
Revised: Two-qubit gates are universal , i.e., every n -qubit gate can be decomposed into a product of gates that act as the identity on n − 2 qubits and as some two-qubit gate on the other 2 qubits.

Original: The gate complexity of an operation refers to the number of two-qubit gates used in a quantum circuit for realizing it.
Revised: The gate complexity of an operation refers to the number of two-qubit gates used in a quantum circuit for realizing it.

Original: Quantum access to a function, referred to as a quantum oracle , must be reversible and allow access to different values of the function in superposition (i.e., for linear combinations of computational basis states).
Revised: Quantum access to a function, referred to as a quantum oracle , must be reversible and allow access to different values of the function in superposition (i.e., for linear combinations of computational basis states).

Original: For example, consider the unitary evaluation oracle O f deﬁned in (1.4).
Revised: For example, consider the unitary evaluation oracle O f deﬁned in (1.4).

Original: Given a probability distribution { p i } ni =1 and a set of points { x i } ni =1 , we have
Revised: Given a probability distribution { p i } ni =1 and a set of points { x i } ni =1 , we have

Original: [Equation]
Revised: [Equation]

Original: Then a measurement would give f ( x i ) with probability p i .
Revised: Then a measurement would give f ( x i ) with probability p i .

Original: However, a quantum oracle can not only simulate random sampling, but can enable uniquely quantum behavior through interference.
Revised: However, a quantum oracle can not only simulate random sampling, but can enable uniquely quantum behavior through interference.

Original: Examples include amplitude ampliﬁcation—the main idea behind Grover’s search algorithm [19] and the amplitude estimation procedure used in this paper—and many other quantum algorithms relying on coherent quantum access to a function.
Revised: Examples include amplitude ampliﬁcation—the main idea behind Grover’s search algorithm [20] and the amplitude estimation procedure used in this paper—and many other quantum algorithms relying on coherent quantum access to a function.

Original: Similar arguments apply to the quantum gradient oracle (1.5).
Revised: Similar arguments apply to the quantum gradient oracle (1.5).

Original: If a classical oracle can be computed by an explicit classical circuit, then the corresponding quantum oracle can be implemented by a quantum circuit of approximately the same size.
Revised: If a classical oracle can be computed by an explicit classical circuit, then the corresponding quantum oracle can be implemented by a quantum circuit of approximately the same size.

Original: Therefore, these quantum oracles provide a useful framework for understanding the quantum complexity of log-concave sampling and normalizing constant estimation.
Revised: Therefore, these quantum oracles provide a useful framework for understanding the quantum complexity of log-concave sampling and normalizing constant estimation.

Original: To sample from a distribution π , it sufﬁces to prepare the state | π (cid:105) :
Revised: To sample from a distribution π , it sufﬁces to prepare the state | π (cid:105) :

Original: = (cid:80) x √ π x | x (cid:105) and then measure it.
Revised: = (cid:80) x √ π x | x (cid:105) and then measure it.

Original: For a Markov chain speciﬁed by a transition matrix P with stationary distribution π , one can construct a corresponding quantum walk operator W ( P ) .
Revised: For a Markov chain speciﬁed by a transition matrix P with stationary distribution π , one can construct a corresponding quantum walk operator W ( P ) .

Original: Intuitively, quantum walks can be viewed as applying a sequence of quantum unitaries on a quantum state encoding the initial distribution to rotate it to the subspace of stationary distribution | π (cid:105) .
Revised: Intuitively, quantum walks can be viewed as applying a sequence of quantum unitaries on a quantum state encoding the initial distribution to rotate it to the subspace of stationary distribution | π (cid:105) .

Original: The number of rotations needed (i.e., the angle between the initial distribution and stationary distribution) depends on the spectral gap of P , and a quantum algorithm can achieve a quadratic speedup via quantum phase estimation and ampliﬁcation algorithms.
Revised: The number of rotations needed (i.e., the angle between the initial distribution and stationary distribution) depends on the spectral gap of P , and a quantum algorithm can achieve a quadratic speedup via quantum phase estimation and ampliﬁcation algorithms.

Original: More background on quantum walk is given in Appendix C.2.2.
Revised: More background on quantum walk is given in Appendix C.2.2.

Original: Notations Throughout the paper, the big-O notations O ( · ) , o ( · ) , Ω( · ) , and Θ( · ) follow common deﬁnitions.
Revised: Notations Throughout the paper, the big-O notations O ( · ) , o ( · ) , Ω( · ) , and Θ( · ) follow common deﬁnitions.

Original: The ˜ O notation omits poly-logarithmic terms, i.e., ˜ O
Revised: The ˜ O notation omits poly-logarithmic terms, i.e., ˜ O

Original: ( f ) :
Revised: ( f ) :

Original: = O ( f poly(log f )) .
Revised: = O ( f poly(log f )) .

Original: We say a function f is L -Lipschitz continuous at x if | f ( x ) − f ( y ) | ≤ L (cid:107) x
Revised: We say a function f is L -Lipschitz continuous at x if | f ( x ) − f ( y ) | ≤ L (cid:107) x

Original: − y (cid:107) for all y sufﬁciently near x .
Revised: − y (cid:107) for all y sufﬁciently near x .

Original: The total variation distance (TV-distance) between two functions f, g : R d → R is deﬁned as
Revised: The total variation distance (TV-distance) between two functions f, g : R d → R is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: Let B ( R d ) denote the Borel σ -ﬁeld of R d .
Revised: Let B ( R d ) denote the Borel σ -ﬁeld of R d .

Original: Given probability measures µ and ν on ( R d , B ( R d )) , a transference plan ζ between µ and ν is deﬁned as a probability measure on ( R d × R d , B ( R d ) ×
Revised: Given probability measures µ and ν on ( R d , B ( R d )) , a transference plan ζ between µ and ν is deﬁned as a probability measure on ( R d × R d , B ( R d ) ×

Original: B ( R d )) such that for any A ⊆ R d , ζ ( A × R d ) = µ ( A ) and ζ ( R d × A ) =
Revised: B ( R d )) such that for any A ⊆ R d , ζ ( A × R d ) = µ ( A ) and ζ ( R d × A ) =

Original: ν ( A ) .
Revised: ν ( A ) .

Original: We let Γ( µ, ν ) denote the set of all transference plans.
Revised: We let Γ( µ, ν ) denote the set of all transference plans.

Original: We let
Revised: We let

Original: [Equation]
Revised: [Equation]

Original: denote the Wasserstein 2-norm between µ and ν .
Revised: denote the Wasserstein 2-norm between µ and ν .

Original:  Quantum Algorithm for Log-Concave Sampling
Revised:  Quantum Algorithm for Log-Concave Sampling

Original: In this section, we describe several quantum algorithms for sampling log-concave distributions.
Revised: In this section, we describe several quantum algorithms for sampling log-concave distributions.

Original: Quantum inexact ULD and ULD-RMM We ﬁrst show that the gradient oracle in the classical
Revised: Quantum inexact ULD and ULD-RMM We ﬁrst show that the gradient oracle in the classical

Original: ULD and ULD-RMM algorithms can be efﬁciently simulated by the quantum evaluation oracle via quantum gradient estimation.
Revised: ULD and ULD-RMM algorithms can be efﬁciently simulated by the quantum evaluation oracle via quantum gradient estimation.

Original: Suppose we are given access to the evaluation oracle (1.4) for f ( x ) .
Revised: Suppose we are given access to the evaluation oracle (1.4) for f ( x ) .

Original: Then by Jordan’s algorithm [23] (see Lemma C.1 for details), there is a quantum algorithm that can compute ∇ f ( x ) with a polynomially small (cid:96) 1 -error by querying the evaluation oracle O (1) times.
Revised: Then by Jordan’s algorithm [24] (see Lemma C.1 for details), there is a quantum algorithm that can compute ∇ f ( x ) with a polynomially small (cid:96) 1 -error by querying the evaluation oracle O (1) times.

Original: Using this, we can prove the following theorem (see Appendix C.1 for details).
Revised: Using this, we can prove the following theorem (see Appendix C.1 for details).

Original: Theorem 3.1 (Informal version of Theorem C.1 and Theorem C.2) .
Revised: Theorem 3.1 (Informal version of Theorem C.1 and Theorem C.2) .

Original: Let ρ ∝ e − f be a d -dimensional log-concave distribution with f satisfying (1.1).
Revised: Let ρ ∝ e − f be a d -dimensional log-concave distribution with f satisfying (1.1).

Original: Given a quantum evaluation oracle for f ,
Revised: Given a quantum evaluation oracle for f ,

Original: • the quantum inexact ULD algorithm uses (cid:101) O ( κ 2 d 1 / 2 (cid:15) − 1 ) queries, and • the quantum inexact ULD-RMM algorithm uses (cid:101)
Revised: • the quantum inexact ULD algorithm uses (cid:101) O ( κ 2 d 1 / 2 (cid:15) − 1 ) queries, and • the quantum inexact ULD-RMM algorithm uses (cid:101)

Original: O ( κ 7 / 6 d 1 / 6 (cid:15)
Revised: O ( κ 7 / 6 d 1 / 6 (cid:15)

Original: − 1 / 3 + κd 1 / 3 (cid:15) − 2 / 3 ) queries, to quantumly sample from a distribution that is (cid:15) -close to ρ in W 2 -distance.
Revised: − 1 / 3 + κd 1 / 3 (cid:15) − 2 / 3 ) queries, to quantumly sample from a distribution that is (cid:15) -close to ρ in W 2 -distance.

Original: We note that the query complexities of our quantum algorithms using a zeroth-order oracle match the state-of-the-art classical ULD [9] and ULD-RMM [37] complexities with a ﬁrst-order oracle.
Revised: We note that the query complexities of our quantum algorithms using a zeroth-order oracle match the state-of-the-art classical ULD [10] and ULD-RMM [37] complexities with a ﬁrst-order oracle.

Original: The main technical difﬁculty of applying the quantum gradient algorithm is that it produces a stochastic gradient oracle in which the output of the quantum algorithm g satisﬁes (cid:107) E [ g ] −∇ f ( x ) (cid:107) 1 ≤ d − Ω(1) .
Revised: The main technical difﬁculty of applying the quantum gradient algorithm is that it produces a stochastic gradient oracle in which the output of the quantum algorithm g satisﬁes (cid:107) E [ g ] −∇ f ( x ) (cid:107) 1 ≤ d − Ω(1) .

Original: In particular, the randomness of the gradient computation is “entangled” with the randomness of the
Revised: In particular, the randomness of the gradient computation is “entangled” with the randomness of the

Original: Markov chain.
Revised: Markov chain.

Original: We use the classical analysis of ULD and ULD-RMM processes [36] to prove that the stochastic gradient will not signiﬁcantly slow down the mixing of ULD processes, and that the error caused by the quantum gradient algorithm can be controlled.
Revised: We use the classical analysis of ULD and ULD-RMM processes [36] to prove that the stochastic gradient will not signiﬁcantly slow down the mixing of ULD processes, and that the error caused by the quantum gradient algorithm can be controlled.

Original: Quantum MALA
Revised: Quantum MALA

Original: We next propose two quantum algorithms with lower query complexity than classical MALA, one with a Gaussian initial distribution and another with a warm-start distribution.
Revised: We next propose two quantum algorithms with lower query complexity than classical MALA, one with a Gaussian initial distribution and another with a warm-start distribution.

Original: The main technical tool we use is a quantum walk in continuous space.
Revised: The main technical tool we use is a quantum walk in continuous space.

Original: The classical MALA (i.e., Metropolized HMC) starts from a Gaussian distribution N (0 , L − 1 I d ) and performs a leapfrog step in each iteration.
Revised: The classical MALA (i.e., Metropolized HMC) starts from a Gaussian distribution N (0 , L − 1 I d ) and performs a leapfrog step in each iteration.

Original: It is well-known that the initial Gaussian state
Revised: It is well-known that the initial Gaussian state

Original: [Equation]
Revised: [Equation]

Original: can be efﬁciently prepared.
Revised: can be efﬁciently prepared.

Original: We show that the quantum walk update operator
Revised: We show that the quantum walk update operator

Original: [Equation]
Revised: [Equation]

Original: can be efﬁciently implemented, where p x → y :=
Revised: can be efﬁciently implemented, where p x → y :=

Original: p ( x, y ) is the transition density from x to y , and the density p satisﬁes (cid:82) R d p ( x, y )
Revised: p ( x, y ) is the transition density from x to y , and the density p satisﬁes (cid:82) R d p ( x, y )

Original: d y = 1 for any x ∈ R d .
Revised: d y = 1 for any x ∈ R d .

Original: Lemma 3.1 (Informal version of Lemma C.6) .
Revised: Lemma 3.1 (Informal version of Lemma C.6) .

Original: The continuous-space quantum walk operator corresponding to the MALA Markov chain can be implemented with O (1) gradient and evaluation queries.
Revised: The continuous-space quantum walk operator corresponding to the MALA Markov chain can be implemented with O (1) gradient and evaluation queries.

Original: In general, it is difﬁcult to quantumly speed up the mixing time of a classical Markov chain, which is upper bounded by O ( δ − 1 log (cid:0) ρ − 1min (cid:1) ) , where δ is the spectral gap.
Revised: In general, it is difﬁcult to quantumly speed up the mixing time of a classical Markov chain, which is upper bounded by O ( δ − 1 log (cid:0) ρ − 1min (cid:1) ) , where δ is the spectral gap.

Original: However, [45] shows that a quadratic speedup is possible when following a sequence of slowly-varying Markov chains.
Revised: However, [45] shows that a quadratic speedup is possible when following a sequence of slowly-varying Markov chains.

Original: More speciﬁcally, let ρ 0 , . .
Revised: More speciﬁcally, let ρ 0 , . .

Original: , ρ r be the stationary distributions of the reversible Markov chains
Revised: , ρ r be the stationary distributions of the reversible Markov chains

Original: M 0 , . .
Revised: M 0 , . .

Original: , M r and let | ρ 0 (cid:105) , . .
Revised: , M r and let | ρ 0 (cid:105) , . .

Original: , | ρ r (cid:105) be the corresponding quantum states.
Revised: , | ρ r (cid:105) be the corresponding quantum states.

Original: Suppose |(cid:104) ρ i |
Revised: Suppose |(cid:104) ρ i |

Original: ρ i +1 (cid:105)| ≥ p for all i ∈ { 0 , . . .
Revised: ρ i +1 (cid:105)| ≥ p for all i ∈ { 0 , . . .

Original: , r − 1 } , and suppose the spectral gaps of M 0 , . . .
Revised: , r − 1 } , and suppose the spectral gaps of M 0 , . . .

Original: , M r are lower-bounded by δ .
Revised: , M r are lower-bounded by δ .

Original: Then we can prepare a quantum state | (cid:101) ρ r (cid:105) that is (cid:15) -close to | ρ r (cid:105) using ˜ O (cid:0) δ − 1 / 2 rp − 1 (cid:1) quantum walk steps.
Revised: Then we can prepare a quantum state | (cid:101) ρ r (cid:105) that is (cid:15) -close to | ρ r (cid:105) using ˜ O (cid:0) δ − 1 / 2 rp − 1 (cid:1) quantum walk steps.

Original: To fulﬁll the slowly-varying condition, we consider an annealing process that goes from ρ 0 = N (0 , L − 1 I d ) to the target distribution ρ M +

Revised: To fulﬁll the slowly-varying condition, we consider an annealing process that goes from ρ 0 = N (0 , L − 1 I d ) to the target distribution ρ M +

Original: = ρ in M = (cid:101) O ( √ d ) stages.
Revised: = ρ in M = (cid:101) O ( √ d ) stages.

Original: At the i th stage, the stationary distribution is ρ i ∝ e − f i with f i := f
Revised: At the i th stage, the stationary distribution is ρ i ∝ e − f i with f i := f

Original: + 12 σ − 2 i (cid:107) x (cid:107) 2 .
Revised: + 12 σ − 2 i (cid:107) x (cid:107) 2 .

Original: By properly choosing σ 1 ≤ · · · ≤ σ M , we prove that this sequence of Markov chains is slowly varying.
Revised: By properly choosing σ 1 ≤ · · · ≤ σ M , we prove that this sequence of Markov chains is slowly varying.

Original: Lemma 3.2 (Informal version of Lemma B.6) .
Revised: Lemma 3.2 (Informal version of Lemma B.6) .

Original: If we take σ 21 = (cid:15) 2 dL
Revised: If we take σ 21 = (cid:15) 2 dL

Original: and σ 2 i +

Revised: and σ 2 i +

Original: =
Revised: =

Original: (1 + 1 √ d ) σ 2 i , then for 0 ≤ i ≤ M , we have |(cid:104) ρ i
Revised: (1 + 1 √ d ) σ 2 i , then for 0 ≤ i ≤ M , we have |(cid:104) ρ i

Original: | ρ
Revised: | ρ

Original: i +1 (cid:105)| ≥ Ω(1) .
Revised: i +1 (cid:105)| ≥ Ω(1) .

Original: Combining Lemma 3.1, Lemma 3.2, andresults on the mixing time of MALA [27], we have:
Revised: Combining Lemma 3.1, Lemma 3.2, and the effective spectral gap of MALA (Lemma C.7), we have:

Original: Theorem 3.2 (Informal version of Theorem C.7) .
Revised: Theorem 3.2 (Informal version of Theorem C.7) .

Original: Let ρ ∝ e − f be a d -dimensional log-concave distribution with f satisfying (1.1) .
Revised: Let ρ ∝ e − f be a d -dimensional log-concave distribution with f satisfying (1.1) .

Original: There is a quantum algorithm
Revised: There is a quantum algorithm

Original: (Algorithm 1) that prepares a state | (cid:101) ρ (cid:105) with (cid:107)| (cid:101) ρ (cid:105) − | ρ (cid:105)(cid:107) ≤ (cid:15) using (cid:101) O ( κ 1 / 2 d ) gradient and evaluation oracle queries. 
Revised: (Algorithm 1) that prepares a state | (cid:101) ρ (cid:105) with (cid:107)| (cid:101) ρ (cid:105) − | ρ (cid:105)(cid:107) ≤ (cid:15) using (cid:101) O ( κ 1 / 2 d ) gradient and evaluation oracle queries. O f i

Revised: ∇ i performing amplitude ampliﬁcation i on the

Original: For the classical MALA with a Gaussian initial distribution, it was shown by [28] that the mixing time is at least (cid:101) Ω( κd ) .
Revised: For the classical MALA with a Gaussian initial distribution, it was shown by [29] that the mixing time is at least (cid:101) Ω( κd ) .

Original: Theorem 3.2 quadratically reduces the κ dependence.
Revised: Theorem 3.2 quadratically reduces the κ dependence.

Original: Note that Algorithm 1 uses a ﬁrst-order oracle, instead of the zeroth-order oracle used in the quantum
Revised: Note that Algorithm 1 uses a ﬁrst-order oracle, instead of the zeroth-order oracle used in the quantum

Original: ULD algorithms.
Revised: ULD algorithms.

Original: The technical barrier to applying the quantum gradient algorithm (Lemma C.1) in the quantum MALA is to analyze the classical MALA with a stochastic gradient oracle.
Revised: The technical barrier to applying the quantum gradient algorithm (Lemma C.1) in the quantum MALA is to analyze the classical MALA with a stochastic gradient oracle.

Original: We currently do not know whether the “entangled randomness” dramatically increases the mixing time.
Revised: We currently do not know whether the “entangled randomness” dramatically increases the mixing time.

Original: More technical details and proofs are provided in Appendix C.
Revised: More technical details and proofs are provided in Appendix C.

Original:  Quantum Algorithm for Estimating Normalizing Constants
Revised:  Quantum Algorithm for Estimating Normalizing Constants

Original: In this section, we apply our quantum log-concave sampling algorithms to the normalizing constant estimation problem.
Revised: In this section, we apply our quantum log-concave sampling algorithms to the normalizing constant estimation problem.

Original: A very natural approach to this problem is via MCMC, which constructs a multi-stage annealing process and uses a sampler at each stage to solve a mean estimation problem.
Revised: A very natural approach to this problem is via MCMC, which constructs a multi-stage annealing process and uses a sampler at each stage to solve a mean estimation problem.

Original: We show how to quantumly speed up these annealing processes and improve the query complexity of estimating normalizing constants.
Revised: We show how to quantumly speed up these annealing processes and improve the query complexity of estimating normalizing constants.

Original: Quantum speedup for the standard annealing process We ﬁrst consider the standard annealing process for log-concave distributions, as already applied in the previous section.
Revised: Quantum speedup for the standard annealing process We ﬁrst consider the standard annealing process for log-concave distributions, as already applied in the previous section.

Original: Recall that we pick parameters σ 1 < σ 2 < · · · < σ M and construct a sequence of Markov chains with stationary distributions
Revised: Recall that we pick parameters σ 1 < σ 2 < · · · < σ M and construct a sequence of Markov chains with stationary distributions

Original: ρ i ∝ e − f i , where f i = f
Revised: ρ i ∝ e − f i , where f i = f

Original: + 12 σ 2 i (cid:107) x (cid:107) 2 .
Revised: + 12 σ 2 i (cid:107) x (cid:107) 2 .

Original: Then, at the i th stage, we estimate the expectation
Revised: Then, at the i th stage, we estimate the expectation

Original: E ρ i [ g i ] where g i
Revised: E ρ i [ g i ] where g i

Original: [Equation]
Revised: [Equation]

Original: exp
Revised: exp

Original: [Equation]
Revised: [Equation]

Original: If we can estimate each expectation with relative error at most O ( (cid:15)/M ) , then the product of these
Revised: If we can estimate each expectation with relative error at most O ( (cid:15)/M ) , then the product of these

Original: M quantities estimates the normalizing constant Z = (cid:82) R d e − f
Revised: M quantities estimates the normalizing constant Z = (cid:82) R d e − f

Original: ( x ) d x with relative error at most (cid:15) .
Revised: ( x ) d x with relative error at most (cid:15) .

Original: For the mean estimation problem, [31] showed that when the relative variance Var ρi
Revised: For the mean estimation problem, [31] showed that when the relative variance Var ρi

Original: [ g i ] E ρi [ g i ] 2 is constant, there is a quantum algorithm for estimating the expectation E ρ i [ g i ] within relative error at most (cid:15) using (cid:101) O
Revised: [ g i ] E ρi [ g i ] 2 is constant, there is a quantum algorithm for estimating the expectation E ρ i [ g i ] within relative error at most (cid:15) using (cid:101) O

Original: (1 /(cid:15) )
Revised: (1 /(cid:15) )

Original: quantum samples from the distribution ρ i .
Revised: quantum samples from the distribution ρ i .

Original: Our annealing schedule satisﬁes the bounded relative variance condition.
Revised: Our annealing schedule satisﬁes the bounded relative variance condition.

Original: Therefore, by the quantum mean estimation algorithm, we improve the sampling complexity of the standard annealing process from (cid:101)
Revised: Therefore, by the quantum mean estimation algorithm, we improve the sampling complexity of the standard annealing process from (cid:101)

Original: O
Revised: O

Original: ( M 2 (cid:15) − 2 ) to (cid:101) O ( M(cid:15) − 1 ) .
Revised: ( M 2 (cid:15) − 2 ) to (cid:101) O ( M(cid:15) − 1 ) .

Original: To further improve the query complexity, we consider using the quantum MALAs developed in the previous section to generate samples.
Revised: To further improve the query complexity, we consider using the quantum MALAs developed in the previous section to generate samples.

Original: Observe that Algorithm 1 outputs a quantum state corresponding to some distribution that is close to ρ i , instead of an individual sample.
Revised: Observe that Algorithm 1 outputs a quantum state corresponding to some distribution that is close to ρ i , instead of an individual sample.

Original: If we can estimate the expectation without destroying the quantum state, then we can reuse the state and evolve it for the ( i + 1) st Markov chain.
Revised: If we can estimate the expectation without destroying the quantum state, then we can reuse the state and evolve it for the ( i + 1) st Markov chain.

Original: Fortunately, we can use non-destructive mean estimation to estimate the expectation and restore the initial states.
Revised: Fortunately, we can use non-destructive mean estimation to estimate the expectation and restore the initial states.

Original: A detailed error analysis of this algorithm can be found in [6, 21].
Revised: A detailed error analysis of this algorithm can be found in [6, 22].

Original: We ﬁrst prepare (cid:101) O
Revised: We ﬁrst prepare (cid:101) O

Original: ( M(cid:15) − 1 ) copies of initial states corresponding to the Gaussian distribution N (0 , L − 1 I d ) .
Revised: ( M(cid:15) − 1 ) copies of initial states corresponding to the Gaussian distribution N (0 , L − 1 I d ) .

Original: Then, for each stage, we apply the non-destructive mean estimation algorithm to estimate the expectation E ρ i [ g i ] and then run quantum MALA to evolve the states |
Revised: Then, for each stage, we apply the non-destructive mean estimation algorithm to estimate the expectation E ρ i [ g i ] and then run quantum MALA to evolve the states |

Original: ρ i (cid:105) to | ρ i +1 (cid:105) .
Revised: ρ i (cid:105) to | ρ i +1 (cid:105) .

Original: This gives our ﬁrst quantum algorithm for estimating normalizing constants.
Revised: This gives our ﬁrst quantum algorithm for estimating normalizing constants.

Original: Theorem 4.1 (Informal version of Theorem D.2) .
Revised: Theorem 4.1 (Informal version of Theorem D.2) .

Original: Let Z be the normalizing constant in (1.3).
Revised: Let Z be the normalizing constant in (1.3).

Original: There is a quantum algorithm
Revised: There is a quantum algorithm

Original: (Algorithm 2) that outputs an estimate (cid:101)
Revised: (Algorithm 2) that outputs an estimate (cid:101)

Original: Z with relative error at most (cid:15) using (cid:101) O (cid:0) d 3 / 2 κ 1 / 2 (cid:15) − 1 (cid:1) queries to the quantum gradient and evaluation oracles.
Revised: Z with relative error at most (cid:15) using (cid:101) O (cid:0) d 3 / 2 κ 1 / 2 (cid:15) − 1 (cid:1) queries to the quantum gradient and evaluation oracles.

Original: Quantum speedup for MLMC Now we consider using multilevel Monte Carlo (MLMC) as the annealing process and show how to achieve quantum speedup.
Revised: Quantum speedup for MLMC Now we consider using multilevel Monte Carlo (MLMC) as the annealing process and show how to achieve quantum speedup.

Original: MLMC was originally developed by [22] for parametric integration; then [16] applied MLMC to simulate stochastic differential equations (SDEs).
Revised: MLMC was originally developed by [23] for parametric integration; then [17] applied MLMC to simulate stochastic differential equations (SDEs).

Original: The idea of MLMC is natural: we choose a different number of samples at each stage based on the cost and variance of that stage.
Revised: The idea of MLMC is natural: we choose a different number of samples at each stage based on the cost and variance of that stage.

Original: To estimate normalizing constants, a variant of MLMC was proposed in [15].
Revised: To estimate normalizing constants, a variant of MLMC was proposed in [16].

Original: Unlike the standard
Revised: Unlike the standard

Original: MLMC for bounding the mean-squared error, they upper bound the bias and the variance separately, and the analysis is technically difﬁcult.
Revised: MLMC for bounding the mean-squared error, they upper bound the bias and the variance separately, and the analysis is technically difﬁcult.

Original: The ﬁrst quantum algorithm based on MLMC was subsequently developed by [2] based on the quantum mean estimation algorithm.
Revised: The ﬁrst quantum algorithm based on MLMC was subsequently developed by [2] based on the quantum mean estimation algorithm.

Original: Roughly speaking, the quantum algorithm can quadratically reduce the (cid:15) -dependence of the sample complexity compared with classical MLMC.
Revised: Roughly speaking, the quantum algorithm can quadratically reduce the (cid:15) -dependence of the sample complexity compared with classical MLMC.

Original: Input: Evaluation oracle O f , gradient oracle O ∇ f
Revised: Input: Evaluation oracle O f , gradient oracle O ∇ f

Original: Output: Estimate (cid:101) Z of Z with relative error at most (cid:15) 1 M ← (cid:101) O ( √ d ) , K ← (cid:101) O ( (cid:15) − 1 ) 2 Compute the cooling schedule parameters σ 1 , . . .
Revised: Output: Estimate (cid:101) Z of Z with relative error at most (cid:15) 1 M ← (cid:101) O ( √ d ) , K ← (cid:101) O ( (cid:15) − 1 ) 2 Compute the cooling schedule parameters σ 1 , . . .

Original: , σ M 3 for j ← 1 , . . .
Revised: , σ M 3 for j ← 1 , . . .

Original: K do
Revised: K do

Original: Prepare the state | ρ 1 ,j (cid:105) ∝ (cid:82) R d e − 14 (cid:107) x (cid:107) 2 /σ 21 | x (cid:105) d x 5 (cid:101)
Revised: Prepare the state | ρ 1 ,j (cid:105) ∝ (cid:82) R d e − 14 (cid:107) x (cid:107) 2 /σ 21 | x (cid:105) d x 5 (cid:101)

Original: Z ← (2 πσ 21 ) d/6 for i ← 1 , . . .
Revised: Z ← (2 πσ 21 ) d/6 for i ← 1 , . . .

Original: , M do (cid:101) g
Revised: , M do (cid:101) g

Original: i ← Non-destructive mean estimation for g i using {| ρ i, 0 (cid:105) , . .
Revised: i ← Non-destructive mean estimation for g i using {| ρ i, 0 (cid:105) , . .

Original: , | ρ i,K (cid:105)} (cid:101) Z ← (cid:101) Z (cid:101) g
Revised: , | ρ i,K (cid:105)} (cid:101) Z ← (cid:101) Z (cid:101) g

Original: i for j ← 1 , . . .
Revised: i for j ← 1 , . . .

Original: , K do | ρ i +1 ,j (cid:105) ← Q UANTUM MALA
Revised: , K do | ρ i +1 ,j (cid:105) ← Q UANTUM MALA

Original: ( O f i +1 , O ∇ ,f i +1 , | ρ i,j (cid:105) )
Revised: ( O f i +1 , O ∇ ,f i +1 , | ρ i,j (cid:105) )

Original: 11 return (cid:101)
Revised: 11 return (cid:101)

Original: Z
Revised: Z

Original: In this work, we apply the quantum accelerated MLMC (QA-MLMC) scheme [2] to simulate underdamped Langevin dynamics as the SDE.
Revised: In this work, we apply the quantum accelerated MLMC (QA-MLMC) scheme [2] to simulate underdamped Langevin dynamics as the SDE.

Original: One challenge in using QA-MLMC is that g i in our setting is not Lipschitz.
Revised: One challenge in using QA-MLMC is that g i in our setting is not Lipschitz.

Original: Fortunately, as suggested by [15], this issue can be resolved by truncating large x and replacing g i by h i :=
Revised: Fortunately, as suggested by [16], this issue can be resolved by truncating large x and replacing g i by h i :=

Original: min (cid:8) g i , exp (cid:0) ( r + i ) 2 σi (1+ α − 1 ) (cid:1)(cid:9) , with the choice α = (cid:101) O (cid:18)
Revised: min (cid:8) g i , exp (cid:0) ( r + i ) 2 σi (1+ α − 1 ) (cid:1)(cid:9) , with the choice α = (cid:101) O (cid:18)

Original: [Equation]
Revised: [Equation]

Original: √
Revised: √

Original: [Equation]
Revised: [Equation]

Original: (cid:19)
Revised: (cid:19)

Original: r
Revised: r

Original: + i =
Revised: + i =

Original: E
Revised: E

Original: ρ i +1 (cid:107) x (cid:107) + Θ( σ i (cid:112) (1 + α ) log(1 /(cid:15) ))
Revised: ρ i +1 (cid:107) x (cid:107) + Θ( σ i (cid:112) (1 + α ) log(1 /(cid:15) ))

Original: (4.2) to ensure h i E ρi
Revised: (4.2) to ensure h i E ρi

Original: g i is O ( σ − 1 i ) Lipschitz.
Revised: g i is O ( σ − 1 i ) Lipschitz.

Original: Furthermore, (cid:12)(cid:12) E
Revised: Furthermore, (cid:12)(cid:12) E

Original: ρ i
Revised: ρ i

Original: ( h i − g i ) (cid:12)(cid:12) <
Revised: ( h i − g i ) (cid:12)(cid:12) <

Original: (cid:15) by Lemmas C.7 and C.8 in [15].
Revised: (cid:15) by Lemmas C.7 and C.8 in [16].

Original: For simplicity, we regard g
Revised: For simplicity, we regard g

Original: i as a Lipschitz continuous function in our main results.
Revised: i as a Lipschitz continuous function in our main results.

Original: We present
Revised: We present

Original: QA-MLMC in Algorithm 3, where the sampling algorithm
Revised: QA-MLMC in Algorithm 3, where the sampling algorithm

Original: A can be chosen to be quantum inexact
Revised: A can be chosen to be quantum inexact

Original: ULD/ULD-RMM or quantum MALA.
Revised: ULD/ULD-RMM or quantum MALA.

Original: O
Revised: O

Original: − 1 ) 2 Compute the initial point x 0 and the step size η3 Compute the number of samples N 1 , . . .
Revised: − 1 ) 2 Compute the initial point x 0 and the step size η3 Compute the number of samples N 1 , . . .

Original: , N K 4 for j ← 1 , . .
Revised: , N K 4 for j ← 1 , . .

Original: K do
Revised: K do

Original: Let η j = η/ 2 j − for i ← 1 , . . .
Revised: Let η j = η/ 2 j − for i ← 1 , . . .

Original: , N j do
Revised: , N j do

Original: Samplei ←
Revised: Sample ←

Original: } i
Revised: } i

Original: and
Revised: and

Original: 9 return (cid:101) R = (cid:101) G 

Revised: 9 return (cid:101) R = (cid:101) G

Original: + (cid:80) Kj
Revised: + (cid:80) Kj

Original: =0 ( (cid:101) G − i − (cid:101) G + i )
Revised: =0 ( (cid:101) G − i − (cid:101) G + i )

Original: This QA-MLMC framework reduces the (cid:15) -dependence of the sampling complexity for estimating normalizing constants from (cid:15) − 2 to (cid:15)
Revised: This QA-MLMC framework reduces the (cid:15) -dependence of the sampling complexity for estimating normalizing constants from (cid:15) − 2 to (cid:15)

Original: −
Revised: −

Original: in both the ULD and ULD-RMM cases, as compared with the state-of-the-art classical results [15].
Revised: in both the ULD and ULD-RMM cases, as compared with the state-of-the-art classical results [16].

Original: Using the quantum inexact ULD and ULD-RMM algorithms (Theorem 3.1) to generate samples, we obtain our second quantum algorithm for estimating normalizing constants (see Appendix D for proofs).
Revised: Using the quantum inexact ULD and ULD-RMM algorithms (Theorem 3.1) to generate samples, we obtain our second quantum algorithm for estimating normalizing constants (see Appendix D for proofs).

Original: Theorem 4.2 (Informal version of Theorem D.3 and Theorem D.4) .
Revised: Theorem 4.2 (Informal version of Theorem D.3 and Theorem D.4) .

Original: Let Z be the normalizing constant in (1.3).
Revised: Let Z be the normalizing constant in (1.3).

Original: There exist quantum algorithms for estimating Z with relative error at most (cid:15) using
Revised: There exist quantum algorithms for estimating Z with relative error at most (cid:15) using

Original: • quantum inexact ULD with (cid:101) O
Revised: • quantum inexact ULD with (cid:101) O

Original: ( d 3 / 2 κ 2 (cid:15) − 1 ) queries to the evaluation oracle
Revised: ( d 3 / 2 κ 2 (cid:15) − 1 ) queries to the evaluation oracle

Original: and
Revised: and

Original: • quantum inexact ULD-RMM with (cid:101) O
Revised: • quantum inexact ULD-RMM with (cid:101) O

Original: (( d 7 / 6 κ 7 / 6 + d 4 / 3 κ ) (cid:15) − 1 ) queries to the evaluation oracle.
Revised: (( d 7 / 6 κ 7 / 6 + d 4 / 3 κ ) (cid:15) − 1 ) queries to the evaluation oracle.

Original:  Quantum Lower Bound
Revised:  Quantum Lower Bound

Original: Finally, we lower bound the quantum query complexity of normalizing constant estimation.
Revised: Finally, we lower bound the quantum query complexity of normalizing constant estimation.

Original: Theorem 5.1.
Revised: Theorem 5.1.

Original: For any ﬁxed positive integer k , given query access (1.4) to a function f : R k
Revised: For any ﬁxed positive integer k , given query access (1.4) to a function f : R k

Original: → R that is 1.5-smooth and 0.5-strongly convex, the quantum query complexity of estimating the partition function Z = (cid:82) R k e − f ( x ) d x within multiplicative error (cid:15) with probability at least 2 / 3 is Ω( (cid:15) − 1 1+4 /k ) .
Revised: → R that is 1.5-smooth and 0.5-strongly convex, the quantum query complexity of estimating the partition function Z = (cid:82) R k e − f ( x ) d x within multiplicative error (cid:15) with probability at least 2 / 3 is Ω( (cid:15) − 1 1+4 /k ) .

Original: The proof of our quantum lower bound is inspired by the construction in Section 5 of [15].
Revised: The proof of our quantum lower bound is inspired by the construction in Section 5 of [16].

Original: They consider a log-concave function whose value is negligible outside a hypercube centered at 0 .
Revised: They consider a log-concave function whose value is negligible outside a hypercube centered at 0 .

Original: The interior of the hypercube is decomposed into cells of two types.
Revised: The interior of the hypercube is decomposed into cells of two types.

Original: The function takes different values on each type, and the normalizing constant estimation problem reduces to determining the number of cells of each type.
Revised: The function takes different values on each type, and the normalizing constant estimation problem reduces to determining the number of cells of each type.

Original: Quantumly, we follow the same construction and reduce the cell counting problem to the Hamming weight problem : given an n -bit Boolean string and two integers (cid:96) <
Revised: Quantumly, we follow the same construction and reduce the cell counting problem to the Hamming weight problem : given an n -bit Boolean string and two integers (cid:96) <

Original: (cid:96) (cid:48) , decide whether the Hamming weight (i.e., the number of ones) of this string is (cid:96) 1 or (cid:96) 2 .
Revised: (cid:96) (cid:48) , decide whether the Hamming weight (i.e., the number of ones) of this string is (cid:96) 1 or (cid:96) 2 .

Original: This problem has a known quantum query lower bound [32], which implies the quantum hardness of estimating the normalizing constant.
Revised: This problem has a known quantum query lower bound [32], which implies the quantum hardness of estimating the normalizing constant.

Original: The full proof of Theorem 5.1 appears in Appendix E.
Revised: The full proof of Theorem 5.1 appears in Appendix E.

Original: Acknowledgements
Revised: Acknowledgements

Original: AMC acknowledges support from the Army Research Ofﬁce (grant W911NF-20-1-0015); the Department of Energy, Ofﬁce of Science, Ofﬁce of Advanced Scientiﬁc Computing Research, Accelerated Research in Quantum Computing program; and the National Science Foundation (grant
Revised: AMC acknowledges support from the Army Research Ofﬁce (grant W911NF-20-1-0015); the Department of Energy, Ofﬁce of Science, Ofﬁce of Advanced Scientiﬁc Computing Research, Accelerated Research in Quantum Computing program; and the National Science Foundation (grant

Original: CCF-1813814).
Revised: CCF-1813814).

Original: TL was supported by a startup fund from Peking University, and the Advanced
Revised: TL was supported by a startup fund from Peking University, and the Advanced

Original: Institute of Information Technology, Peking University.
Revised: Institute of Information Technology, Peking University.

Original: JPL was supported by the National Science
Revised: JPL was supported by the National Science

Original: Foundation (grant CCF-1813814), an NSF Quantum Information Science and Engineering Network (QISE-NET) triplet award (DMR-1747426), a Simons Foundation award (No. 825053), and the Simons Quantum Postdoctoral Fellowship.
Revised: Foundation (grant CCF-1813814), an NSF Quantum Information Science and Engineering Network (QISE-NET) triplet award (DMR-1747426), a Simons Foundation award (No. 825053), and the Simons Quantum Postdoctoral Fellowship.

Original: RZ was supported by the University Graduate Continuing
Revised: RZ was supported by the University Graduate Continuing

Original: Fellowship from UT Austin.
Revised: Fellowship from UT Austin.