arXiv subjects

Jiaqing Jiang

Publications and source records attributed to Jiaqing Jiang.

18 recordsLinked to original sources

arXiv2026

Gibbs state preparation for commuting Hamiltonian: Mapping to classical Gibbs sampling

Gibbs state preparation, or Gibbs sampling, is a key computational technique extensively used in physics, statistics, and other scientific fields. Recent efforts for designing fast mixing Gibbs samplers for quantum Hamiltonians have largely focused on commuting local Hamiltonians (CLHs), a non-trivial subclass of Hamiltonians which include highly entangled systems such as the Toric code and quantum double model. Most previous Gibbs samplers relied on simulating the Davies generator, which is a Lindbladian associated with the thermalization process in nature. Instead of using the Davies generator, we design a different Gibbs sampler for various CLHs by giving a reduction to classical Hamiltonians, in the sense that one can efficiently prepare the Gibbs state for some CLH $H$ on a quantum computer as long as one can efficiently do classical Gibbs sampling for the corresponding classical Hamiltonian $H^{(c)}$. We demonstrate that our Gibbs sampler is able to replicate state-of-the-art results as well as prepare the Gibbs state in regimes which were previously unknown, such as the low temperature region, as long as there exists fast mixing Gibbs samplers for the corresponding classical Hamiltonians. Our reductions are as follows. - If $H$ is a 2-local qudit CLH, then $H^{(c)}$ is a 2-local qudit classical Hamiltonian. - If $H$ is a 4-local qubit CLH on 2D lattice and there are no classical qubits, then $H^{(c)}$ is a 2-local qudit classical Hamiltonian on a planar graph. As an example, our algorithm can prepare the Gibbs state for the (defected) Toric code at any non-zero temperature in $O(n^2 poly(log n))$ time. - If $H$ is a 4-local qubit CLH on 2D lattice and there are classical qubits, assuming that quantum terms are uniformly correctable, then $H^{(c)}$ is a constant-local classical Hamiltonian.

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arXiv2026

Thermal expectation estimation via single-trajectory Gibbs sampling with non-destructive measurements

Estimating thermal expectation values of quantum many-body systems is a central challenge in physics, chemistry, and materials science. Standard quantum Gibbs sampling protocols address this task by preparing the Gibbs state from scratch after every measurement, incurring a full mixing time cost at each step. Recent advances in single-trajectory Gibbs sampling [Jiang et al. 2026] substantially reduce this overhead: once stationarity is reached, measurements can be collected along a single trajectory without re-thermalizing, provided the measurement channel preserves the Gibbs ensemble. However, explicit constructions of such non-destructive measurements have been limited primarily to observables that commute with the Hamiltonian. In this work, we fundamentally extend the single-trajectory framework to arbitrary, non-commuting observables. We provide two measurement constructions that extract measurement information without fully destroying the Gibbs state, thereby eliminating the need for full re-mixing between samples. First, we construct a measurement that satisfies exact detailed balance. This ensures the system remains in equilibrium throughout the trajectory, allowing measurement outcomes to decorrelate in an autocorrelation time that could be significantly shorter than the global mixing time. Second, assuming the underlying quantum Gibbs sampler has a positive spectral gap, we design a simplified measurement scheme that ensures the post-selected state serves as a warm start for rapid re-mixing. This approach successfully decouples the resampling cost from the global mixing time. Both measurement schemes admit efficient quantum circuit implementations, requiring only polylogarithmic Hamiltonian simulation time.

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arXiv2026

Accelerating quantum Gibbs sampling without quantum walks

Szegedy's quantum walk gives a generic quadratic speedup for reversible classical Markov chains, but extending this mechanism to quantum Gibbs sampling has remained challenging beyond special cases. We present a walk-free quantum algorithm for preparing purified Gibbs states with a quadratic improvement in spectral-gap dependence for a broad class of quantum Gibbs samplers that satisfy exact Kubo-Martin-Schwinger detailed balance. Our main structural result is an explicit factorization of the corresponding parent Hamiltonian into noncommutative first-order operators. This turns purified Gibbs-state preparation into a singular-value filtering problem and enables a quantum singular value transformation algorithm with quadratically improved gap dependence under standard coherent-access assumptions. The framework applies to several efficiently implementable Gibbs samplers beyond the Davies setting. We also introduce an auxiliary dissipative dynamics based on the same factorization, which can be used to generate warm starts in the doubled Hilbert space in metastable regimes.

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arXiv2026

Predicting properties of quantum thermal states from a single trajectory

Estimating thermal expectation values of observables is a fundamental task in quantum physics, quantum chemistry, and materials science. While recent quantum algorithms have enabled efficient quantum preparation of thermal states, observable estimation via sampling remains costly: a straightforward implementation separates successive measurements by a full mixing time in order to ensure samples are approximately independent. In this work, we show that the sampling cost can be substantially reduced by using a single Gibbs-sampling trajectory. After a single burn-in period, we interleave coherent measurements that satisfy detailed balance with respect to the target Gibbs state. The efficiency of this approach rests on the fact that, in many settings, the autocorrelation time can be significantly shorter than the mixing time. For energy estimation (and more generally for observables commuting with the Hamiltonian), we implement the required measurements using Gaussian-filtered quantum phase estimation with only logarithmic overhead. We also introduce a weighted operator Fourier transform technique to mitigate measurement-induced disturbance for general observables.

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arXiv2026

High-Temperature Fermionic Gibbs States are Mixtures of Gaussian States

Efficient simulation of a quantum system generally relies on structural properties of the quantum state. Motivated by the recent results by Bakshi et al. on the sudden death of entanglement in high-temperature Gibbs states of quantum spin systems, we study the high-temperature Gibbs states of bounded-degree local fermionic Hamiltonians, which include the special case of geometrically local fermionic systems. We prove that at a sufficiently high temperature that is independent of the system size, the Gibbs state is a probabilistic mixture of fermionic Gaussian states. This forms the basis of an efficient classical algorithm to prepare the Gibbs state by sampling from a distribution of fermionic Gaussian states. As a contrasting example, we show that high-temperature Gibbs states of the Sachdev-Ye-Kitaev (SYK) model are not convex mixtures of Gaussian states.

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arXiv2025

Convergence of the Cumulant Expansion and Polynomial-Time Algorithm for Weakly Interacting Fermions

We propose a randomized algorithm to compute the log-partition function of weakly interacting fermions with polynomial runtime in both the system size and precision. Although weakly interacting fermionic systems are considered tractable for many computational methods such as the diagrammatic quantum Monte Carlo, a mathematically rigorous proof of polynomial runtime has been lacking. In this work we first extend the proof techniques developed in previous works for proving the convergence of the cumulant expansion in periodic systems to the non-periodic case. A key equation used to analyze the sum of connected Feynman diagrams, which we call the tree-determinant expansion, reveals an underlying tree structure in the summation. This enables us to design a new randomized algorithm to compute the log-partition function through importance sampling augmented by belief propagation. This approach differs from the traditional method based on Markov chain Monte Carlo, whose efficiency is hard to guarantee, and enables us to obtain a algorithm with provable polynomial runtime.

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arXiv2025

Local Hamiltonian Problem with succinct ground state is MA-Complete

Finding the ground energy of a quantum system is a fundamental problem in condensed matter physics and quantum chemistry. Existing classical algorithms for tackling this problem often assume that the ground state has a succinct classical description, i.e. a poly-size classical circuit for computing the amplitude. Notable examples of succinct states encompass matrix product states, contractible projected entangled pair states, and states that can be represented by classical neural networks. We study the complexity of the local Hamiltonian problem with succinct ground state. We prove this problem is MA-Complete. The Hamiltonian we consider is general and might not be stoquastic. The MA verification protocol is based on the fixed node quantum Monte Carlo method, particularly the variant of the continuous-time Markov chain introduced by Bravyi et.al. [BCGL22]. Based on our work, we also introduce a notion of strong guided states, and conjecture that the local Hamiltonian problem with strong guided state is MA-Complete, which will be in contrast with the QCMA-Complete result of the local Hamiltonian problem with standard guided states [WFC23,GLG22].

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arXiv2025

Sign problem in tensor network contraction

We investigate how the computational difficulty of contracting tensor networks depends on the sign structure of the tensor entries. Using results from computational complexity, we observe that the approximate contraction of tensor networks with only positive entries has lower complexity. This raises the question how this transition in computational complexity manifests itself in the hardness of different contraction schemes. We pursue this question by studying random tensor networks with varying bias towards positive entries. First, we consider contraction via Monte Carlo sampling, and find that the transition from hard to easy occurs when the entries become predominantly positive; this can be seen as a tensor network manifestation of the Quantum Monte Carlo sign problem. Second, we analyze the commonly used contraction based on boundary tensor networks. Its performance is governed by the amount of correlations (entanglement) in the tensor network. Remarkably, we find that the transition from hard to easy (i.e., from a volume law to a boundary law scaling of entanglement) occurs already for a slight bias towards a positive mean, and the earlier the larger the bond dimension is. This is in contrast to both expectations and the behavior found in Monte Carlo contraction. We gain further insight into this early transition from the study of an effective statmech model. Finally, we investigate the computational difficulty of computing expectation values of tensor network wavefunctions, i.e., PEPS, where we find that the complexity of entanglement-based contraction always remains low. We explain this by providing a local transformation which maps PEPS expectation values to a positive-valued tensor network. This not only provides insight into the origin of the observed boundary law entanglement scaling, but also suggests new approaches towards PEPS contraction based on positive decompositions.

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arXiv2024

Positive bias makes tensor-network contraction tractable

Tensor network contraction is a powerful computational tool in quantum many-body physics, quantum information and quantum chemistry. The complexity of contracting a tensor network is thought to mainly depend on its entanglement properties, as reflected by the Schmidt rank across bipartite cuts. Here, we study how the complexity of tensor-network contraction depends on a different notion of quantumness, namely, the sign structure of its entries. We tackle this question rigorously by investigating the complexity of contracting tensor networks whose entries have a positive bias. We show that for intermediate bond dimension d>~n, a small positive mean value >~1/d of the tensor entries already dramatically decreases the computational complexity of approximately contracting random tensor networks, enabling a quasi-polynomial time algorithm for arbitrary 1/poly(n) multiplicative approximation. At the same time exactly contracting such tensor networks remains #P-hard, like for the zero-mean case [HHEG20]. The mean value 1/d matches the phase transition point observed in [CJHS24]. Our proof makes use of Barvinok's method for approximate counting and the technique of mapping random instances to statistical mechanical models. We further consider the worst-case complexity of approximate contraction of positive tensor networks, where all entries are non-negative. We first give a simple proof showing that a multiplicative approximation with error exponentially close to one is at least StoqMA-hard. We then show that when considering additive error in the matrix 1-norm, the contraction of positive tensor network is BPP-Complete. This result compares to Arad and Landau's [AL10] result, which shows that for general tensor networks, approximate contraction up to matrix 2-norm additive error is BQP-Complete.

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arXiv2024

Quantum Metropolis Sampling via Weak Measurement

Gibbs sampling is a crucial computational technique used in physics, statistics, and many other scientific fields. For classical Hamiltonians, the most commonly used Gibbs sampler is the Metropolis algorithm, known for having the Gibbs state as its unique fixed point. For quantum Hamiltonians, designing provably correct Gibbs samplers has been more challenging. [TOV+11] introduced a novel method that uses quantum phase estimation (QPE) and the Marriot-Watrous rewinding technique to mimic the classical Metropolis algorithm for quantum Hamiltonians. The analysis of their algorithm relies upon the use of a boosted and shift-invariant version of QPE which may not exist [CKBG23]. Recent efforts to design quantum Gibbs samplers take a very different approach and are based on simulating Davies generators [CKBG23,CKG23,RWW23,DLL24]. Currently, these are the only provably correct Gibbs samplers for quantum Hamiltonians. We revisit the inspiration for the Metropolis-style algorithm of [TOV+11] and incorporate weak measurement to design a conceptually simple and provably correct quantum Gibbs sampler, with the Gibbs state as its approximate unique fixed point. Our method uses a Boosted QPE which takes the median of multiple runs of QPE, but we do not require the shift-invariant property. In addition, we do not use the Marriott-Watrous rewinding technique which simplifies the algorithm significantly.

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arXiv2023

Commuting Local Hamiltonian Problem on 2D beyond qubits

We study the complexity of local Hamiltonians in which the terms pairwise commute. Commuting local Hamiltonians (CLHs) provide a way to study the role of non-commutativity in the complexity of quantum systems and touch on many fundamental aspects of quantum computing and many-body systems, such as the quantum PCP conjecture and the area law. Despite intense research activity since Bravyi and Vyalyi introduced the CLH problem two decades ago [BV03], its complexity remains largely unresolved; it is only known to lie in NP for a few special cases. Much of the recent research has focused on the physically motivated 2D case, where particles are located on vertices of a 2D grid and each term acts non-trivially only on the particles on a single square (or plaquette) in the lattice. In particular, Schuch [Sch11] showed that the CLH problem on 2D with qubits is in NP. Aharonov, Kenneth and Vigdorovich~[AKV18] then gave a constructive version of this result, showing an explicit algorithm to construct a ground state. Resolving the complexity of the 2D CLH problem with higher dimensional particles has been elusive. We prove two results for the CLH problem in 2D: (1) We give a non-constructive proof that the CLH problem in 2D with qutrits is in NP. As far as we know, this is the first result for the commuting local Hamiltonian problem on 2D beyond qubits. Our key lemma works for general qudits and might give new insights for tackling the general case. (2) We consider the factorized case, also studied in [BV03], where each term is a tensor product of single-particle Hermitian operators. We show that a factorized CLH in 2D, even on particles of arbitrary finite dimension, is equivalent to a direct sum of qubit stabilizer Hamiltonians. This implies that the factorized 2D CLH problem is in NP. This class of CLHs contains the Toric code as an example.

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arXiv2023

Lower bound for the T count via unitary stabilizer nullity

We introduce magic measures to quantify the nonstabilizerness of multiqubit quantum gates and establish lower bounds on the $T$ count for fault-tolerant quantum computation. First, we introduce the stabilizer nullity of multi-qubit unitary, which is based on the subgroup of the quotient Pauli group associated with the unitary. This unitary stabilizer nullity extends the state-stabilizer nullity by Beverland et al. to a dynamic version. In particular, we show this nonstabilizerness measure has desirable properties such as subadditivity under composition and additivity under tensor product. Second, we prove that a given unitary's stabilizer nullity is a lower bound for the $T$ count, utilizing the above properties in gate synthesis. Third, we compare the state- and the unitary-stabilizer nullity, proving that the lower bounds for the $T$ count obtained by the unitary-stabilizer nullity are never less than the state-stabilizer nullity. Moreover, we show an explicit $n$-qubit unitary family of unitary-stabilizer nullity $2n$, which implies that its $T$ count is at least $2n$. This gives an example where the bounds derived by the unitary-stabilizer nullity strictly outperform the state-stabilizer nullity by a factor of $2$. We finally showcase the advantages of unitary-stabilizer nullity in estimating the $T$ count of quantum gates with interests.

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arXiv2022

Optimal Space-Depth Trade-Off of CNOT Circuits in Quantum Logic Synthesis

Decoherence -- in the current physical implementations of quantum computers -- makes depth reduction a vital task in quantum-circuit design. Moore and Nilsson (SIAM Journal of Computing, 2001) demonstrated that additional qubits -- known as ancillae -- can be used to provide an extended space to parallelize quantum circuits. Specifically, they proved that, with $O(n^2)$ ancillae, any $n$-qubit CNOT circuit can be transformed into an equivalent one of $O(\log n)$ depth. However, the near-term quantum technologies can only support a limited amount of qubits, making space-depth trade-off a fundamental research subject for quantum-circuit synthesis. In this work, we establish an asymptotically optimal space-depth trade-off for CNOT circuits. We prove that any $n$-qubit CNOT circuit can be parallelized to $$O\left(\max\left\{\log n, \frac{n^2}{(n+m)\log (n+m)}\right\}\right)$$ depth with $m$ ancillae. This bound is tight even if the task is expanded from exact synthesis to the approximation of CNOT circuits with arbitrary two-qubit quantum gates. Our result can be extended to stabilizer circuits via the reduction by Aaronson and Gottesman (Physical Review A, 2004). Furthermore, we provide hardness evidence for optimizing CNOT circuits in terms of size or depth. Our result has improved upon two previous papers that motivated our work. Moore-Nilsson's construction (aforementioned) for $O(\log n)$-depth CNOT circuit synthesis: We have reduced their need for ancillae by a factor of $\log^2 n$ by showing that $m= O(n^2/\log^2 n)$ additional qubits -- which is asymptotically optimal -- suffice to build equivalent $O(\log n)$-depth $O(n^2/\log n)$-size CNOT circuits. Patel-Markov-Hayes's construction (Quantum Information & Computation 2008) for $m = 0$: We have reduced their depth by a factor of $n$ and achieved the asymptotically optimal bound of $O(n/\log n)$.

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arXiv2021

Physical Implementability of Linear Maps and Its Application in Error Mitigation

Completely positive and trace-preserving maps characterize physically implementable quantum operations. On the other hand, general linear maps, such as positive but not completely positive maps, which can not be physically implemented, are fundamental ingredients in quantum information, both in theoretical and practical perspectives. This raises the question of how well one can simulate or approximate the action of a general linear map by physically implementable operations. In this work, we introduce a systematic framework to resolve this task using the quasiprobability decomposition technique. We decompose a target linear map into a linear combination of physically implementable operations and introduce the physical implementability measure as the least amount of negative portion that the quasiprobability must pertain, which directly quantifies the cost of simulating a given map using physically implementable quantum operations. We show this measure is efficiently computable by semidefinite programs and prove several properties of this measure, such as faithfulness, additivity, and unitary invariance. We derive lower and upper bounds in terms of the Choi operator's trace norm and obtain analytic expressions for several linear maps of practical interests. Furthermore, we endow this measure with an operational meaning within the quantum error mitigation scenario: it establishes the lower bound of the sampling cost achievable via the quasiprobability decomposition technique. In particular, for parallel quantum noises, we show that global error mitigation has no advantage over local error mitigation.

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arXiv2021

General Quantum Bernoulli Factory: Framework Analysis and Experiments

The unremitting pursuit for quantum advantages gives rise to the discovery of a quantum-enhanced randomness processing named quantum Bernoulli factory (QBF). This quantum enhanced process can show its priority over the corresponding classical process through readily available experimental resources, thus in the near term it may be capable of accelerating the applications of classical Bernoulli factories, such as the widely used sampling algorithms. In this work, we provide the framework analysis of the QBF. We thoroughly analyze the quantum state evolution in this process, discovering the field structure of the constructible quantum states. Our framework analysis shows that naturally, the previous works can be described as specific instances of this framework. Then, as a proof of principle, we experimentally demonstrate this framework via an entangled two-photon source along with a reconfigurable photonic logic, and show the advantages of the QBF over the classical model through a classically infeasible instance. These results may stimulate the discovery of advantages of the quantum randomness processing in a wider range of tasks, as well as its potential applications.

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arXiv2019

Structured decomposition for reversible Boolean functions

Reversible Boolean function is a one-to-one function which maps $n$-bit input to $n$-bit output. Reversible logic synthesis has been widely studied due to its relationship with low-energy computation as well as quantum computation. In this work, we give a structured decomposition for even reversible Boolean functions (RBF). Specifically, for $n\geq 6$, any even $n$-bit RBF can be decomposed to $7$ blocks of $(n-1)$-bit RBF, where $7$ is a constant independent of $n$; and the positions of those blocks have large degree of freedom. Moreover, if the $(n-1)$-bit RBFs are required to be even as well, we show for $n\geq 10$, $n$-bit RBF can be decomposed to $10$ even $(n-1)$-bit RBFs. For simplicity, we say our decomposition has block depth $7$ and even block depth $10$. Our result improves Selinger's work in block depth model, by reducing the constant from $9$ to $7$; and from $13$ to $10$ when the blocks are limited to be even. We emphasize that our setting is a bit different from Selinger's. In Selinger's constructive proof, each block is one of two specific positions and thus the decomposition has an alternating structure. We relax this restriction and allow each block to act on arbitrary $(n-1)$ bits. This relaxation keeps the block structure and provides more candidates when choosing positions of blocks.

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arXiv2018

Local unitary classification for sets of generalized Bell states

In this paper, we study the local unitary classification for pairs (triples) of generalized Bell states, based on the local unitary equivalence of two sets. In detail, we firstly introduce some general unitary operators which give us more local unitary equivalent sets besides Clifford operators. And then we present two necessary conditions for local unitary equivalent sets which can be used to examine the local inequivalence. Following this approach, we completely classify all of pairs in $d\otimes d$ quantum system into $\prod_{j=1}^{n} (k_{j} + 1) $ LU-inequivalent pairs when the prime factorization of $d=\prod_{j=1}^{n}p_j^{k_j}$. Moreover, all of triples in $p^α\otimes p^α$ quantum system for prime $p$ can be partitioned into $\frac{(α+ 3)}{6}p^α + O(αp^{α-1})$ LU-inequivalent triples, especially, when $α=2$ and $p>2$, there are exactly $\lfloor \frac{5}{6}p^{2}\rfloor + \lfloor \frac{p-2}{6}+(-1)^{\lfloor\frac{p}{3}\rfloor}\frac{p}{3}\rfloor + 3$ LU-inequivalent triples.

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arXiv2018

Quantum-to-Quantum Bernoulli Factory

Given a coin with unknown bias $p\in [0,1]$, can we exactly simulate another coin with bias $f(p)$? The exact set of simulable functions has been well characterized 20 years ago. In this paper, we ask the quantum counterpart of this question: Given the quantum coin $|p\rangle=\sqrt{p}|0\rangle+\sqrt{1-p}|1\rangle$, can we exactly simulate another quantum coin $|f(p)\rangle=\sqrt{f(p)}|0\rangle+\sqrt{1-f(p)}|1\rangle$? We give the full characterization of simulable quantum state $k_0(p)|0\rangle+k_1(p)|1\rangle$ from quantum coin $|p\rangle=\sqrt{p}|0\rangle+\sqrt{1-p}|1\rangle$, and present an algorithm to transform it. Surprisingly, we show that simulable sets in the quantum-to-quantum case and classical-to-classical case have no inclusion relationship with each other.

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