We study the structure of the ground states of local stoquastic Hamiltonians and show that under mild assumptions the following distributions can efficiently approximate one another: (a) distributions arising from ground states of stoquastic Hamiltonians, (b) distributions arising from ground states of stoquastic frustration-free Hamiltonians, (c) Gibbs distributions of local classical Hamiltonian, and (d) distributions represented by real-valued Deep Boltzmann machines. In addition, we highlight regimes where it is possible to efficiently classically sample from the above distributions.
We propose a two qubit experiment for validating tunable antiferromagnetic $XX$ interactions in quantum annealing. Such interactions allow the time-dependent Hamiltonian to be non-stoquastic, and the instantaneous ground state can have negative amplitudes in the computational basis. Our construction relies on how the degeneracy of the Ising Hamiltonian's ground states is broken away from the end point of the anneal: above a certain value of the antiferromagnetic $XX$ interaction strength, the perturbative ground state at the end of the anneal changes from a symmetric to an antisymmetric state.…
Stoquastic Hamiltonians are characterized by the property that their off-diagonal matrix elements in the standard product basis are real and non-positive. Many interesting quantum models fall into this class including the Transverse field Ising Model (TIM), the Heisenberg model on bipartite graphs, and the bosonic Hubbard model. Here we consider the problem of estimating the ground state energy of a local stoquastic Hamiltonian $H$ with a promise that the ground state of $H$ has a non-negligible correlation with some `guiding' state that admits a concise classical description. A formalized ver…
We study several problems related to properties of non-negative matrices that arise at the boundary between quantum and classical probabilistic computation. Our results are twofold. First, we identify a large class of quantum Hamiltonians describing systems of qubits for which the adiabatic evolution can be efficiently simulated on a classical probabilistic computer. These are stoquastic local Hamiltonians with a "frustration free" ground-state. A Hamiltonian belongs to this class iff it can be represented as $H=\sum_a H_a$ where (1) every term $H_a$ acts non-trivially on a constant number of …
We show that finding the lowest eigenvalue of a 3-local symmetric stochastic matrix is QMA-complete. We also show that finding the highest energy of a stoquastic Hamiltonian is QMA-complete and that adiabatic quantum computation using certain excited states of a stoquastic Hamiltonian is universal. We also show that adiabatic evolution in the ground state of a stochastic frustration free Hamiltonian is universal. Our results give a new QMA-complete problem arising in the classical setting of Markov chains, and new adiabatically universal Hamiltonians that arise in many physical systems.
Variational Monte Carlo with neural network quantum states has proven to be a promising avenue for evaluating the ground state energy of spin Hamiltonians. However, despite continuous efforts the performance of the method on frustrated Hamiltonians remains significantly worse than those on stoquastic Hamiltonians that are sign-free. We present a detailed and systematic study of restricted Boltzmann machine (RBM) based variational Monte Carlo for quantum spin chains, resolving how relevant stoquasticity is in this setting. We show that in most cases, when the Hamiltonian is phase connected with…
The Local Hamiltonian problem (finding the ground state energy of a quantum system) is known to be QMA-complete. The Local Consistency problem (deciding whether descriptions of small pieces of a quantum system are consistent) is also known to be QMA-complete. Here we consider special cases of Local Hamiltonian, for ``stoquastic'' and 1-dimensional systems, that seem to be strictly easier than QMA. We show that there exist analogous special cases of Local Consistency, that have equivalent complexity (up to poly-time oracle reductions). Our main technical tool is a new reduction from Local Consi…
Quantum fluctuations driven by non-stoquastic Hamiltonians have been conjectured to be an important and perhaps essential missing ingredient for achieving a quantum advantage with adiabatic optimization. We introduce a transformation that maps every non-stoquastic adiabatic path ending in a classical Hamiltonian to a corresponding stoquastic adiabatic path by appropriately adjusting the phase of each matrix entry in the computational basis. We compare the spectral gaps of these adiabatic paths and find both theoretically and numerically that the paths based on non-stoquastic Hamiltonians have …
We have proposed the energy landscape transformation of Ising problems (ELTIP), which changes the combination of the state and eigenvalue without changing all the original eigenvalues [arXiv:2202.05927]. We study how the ELTIP affects the anti-crossing between two levels of the ground and first excited states during quantum annealing. We use a 5-spin maximum-weighted independent set for the problem to numerically investigate the anticrossing. For comparison, we introduce a non-stoquastic Hamiltonian that adds antiferromagnetic interaction to the normal transverse magnetic field. Annealing with…
One of the distinct features of quantum mechanics is that the probability amplitude can have both positive and negative signs, which has no classical counterpart as the classical probability must be positive. Consequently, one possible way to achieve quantum speedup is to explicitly harness this feature. Unlike a stoquastic Hamiltonian whose ground state has only positive amplitudes (with respect to the computational basis), a non-stoquastic Hamiltonian can be eventually stoquastic or properly non-stoquastic when its ground state has both positive and negative amplitudes. In this paper, we des…
Quantum annealing (QA) is a promising method for solving combinatorial optimization problems whose solutions are embedded into a ground state of the Ising Hamiltonian. This method employs two types of Hamiltonians: a driver Hamiltonian and a problem Hamiltonian. After a sufficiently slow change from the driver Hamiltonian to the problem Hamiltonian, we can obtain the target ground state that corresponds to the solution. The inclusion of non-stoquastic terms in the driver Hamiltonian is believed to enhance the efficiency of the QA. Meanwhile, decoherence is regarded as of the main obstacles for…
We study the energy gap between the ground state and the first excited state of a mean-field-type non-stoquastic Hamiltonian by a semi-classical analysis. The fully connected mean-field model with $p$-body ferromagnetic interactions under a transverse field has a first-order quantum phase transition for $p\ge 3$. This first-order transition is known to be reduced to second order for $p\ge 5$ by an introduction of antiferromagnetic transverse interactions, which makes the Hamiltonian non-stoquastic. This reduction of the order of transition means an exponential speedup of quantum annealing by a…
What could happen if we pinned a single qubit of a system and fixed it in a particular state? First, we show that this can greatly increase the complexity of static questions -- ground state properties of local Hamiltonian problems with restricted types of terms. In particular, we show that the Pinned commuting and Pinned Stoquastic Local Hamiltonian problems are QMA complete. Second, we show that pinning a single qubit via often repeated measurements also results in universal quantum computation already with commuting and stoquastic Hamiltonians. Finally, we discuss variants of the Ground Sta…
We study complexity of several problems related to the Transverse field Ising Model (TIM). First, we consider the problem of estimating the ground state energy known as the Local Hamiltonian Problem (LHP). It is shown that the LHP for TIM on degree-3 graphs is equivalent modulo polynomial reductions to the LHP for general k-local `stoquastic' Hamiltonians with any constant $k\ge 2$. This result implies that estimating the ground state energy of TIM on degree-3 graphs is a complete problem for the complexity class StoqMA - an extension of the classical class MA. As a corollary, we complete the …
Strongly interacting quantum systems described by non-stoquastic Hamiltonians exhibit rich low-temperature physics. Yet, their study poses a formidable challenge, even for state-of-the-art numerical techniques. Here, we investigate systematically the performance of a class of universal variational wave-functions based on artificial neural networks, by considering the frustrated spin-$1/2$ $J_1-J_2$ Heisenberg model on the square lattice. Focusing on neural network architectures without physics-informed input, we argue in favor of using an ansatz consisting of two decoupled real-valued networks…
Most experimental and theoretical studies of adiabatic optimization use stoquastic Hamiltonians, whose ground states are expressible using only real nonnegative amplitudes. This raises a question as to whether classical Monte Carlo methods can simulate stoquastic adiabatic algorithms with polynomial overhead. Here, we analyze diffusion Monte Carlo algorithms. We argue that, based on differences between L1 and L2 normalized states, these algorithms suffer from certain obstructions preventing them from efficiently simulating stoquastic adiabatic evolution in generality. In practice however, we o…
The derandomization of MA, the probabilistic version of NP, is a long standing open question. In this work, we connect this problem to a variant of another major problem: the quantum PCP conjecture. Our connection goes through the surprising quantum characterization of MA by Bravyi and Terhal. They proved the MA-completeness of the problem of deciding whether the groundenergy of a uniform stoquastic local Hamiltonian is zero or inverse polynomial. We show that the gapped version of this problem, i.e. deciding if a given uniform stoquastic local Hamiltonian is frustration-free or has energy at …
We investigate the relationship between the energy spectrum of a local Hamiltonian and the geometric properties of its ground state. By generalizing a standard framework from the analysis of Markov chains to arbitrary (non-stoquastic) Hamiltonians we are naturally led to see that the spectral gap can always be upper bounded by an isoperimetric ratio that depends only on the ground state probability distribution and the range of the terms in the Hamiltonian, but not on any other details of the interaction couplings. This means that for a given probability distribution the inequality constrains …
The debate around the potential superiority of quantum annealers over their classical counterparts has been ongoing since the inception of the field by Kadowaki and Nishimori close to two decades ago. Recent technological breakthroughs in the field, which have led to the manufacture of experimental prototypes of quantum annealing optimizers with sizes approaching the practical regime, have reignited this discussion. However, the demonstration of quantum annealing speedups remains to this day an elusive albeit coveted goal. Here, we examine the power of quantum annealers to provide a different …
Path integral quantum Monte Carlo (PIMC) is a method for estimating thermal equilibrium properties of stoquastic quantum spin systems by sampling from a classical Gibbs distribution using Markov chain Monte Carlo. The PIMC method has been widely used to study the physics of materials and for simulated quantum annealing, but these successful applications are rarely accompanied by formal proofs that the Markov chains underlying PIMC rapidly converge to the desired equilibrium distribution. In this work we analyze the mixing time of PIMC for 1D stoquastic Hamiltonians, including disordered transv…
Cheeger inequalities bound the spectral gap $γ$ of a space by isoperimetric properties of that space and vice versa. In this paper, I derive Cheeger-type inequalities for nonpositive matrices (aka stoquastic Hamiltonians), real matrices, and Hermitian matrices. For matrices written $H = L+W$, where $L$ is either a combinatorial or normalized graph Laplacian, I show that: (1) when $W$ is diagonal and $L$ has maximum degree $d_{\max}$, $2h \geq γ\geq \sqrt{h^2 + d_{\max}^2}-d_\max$; (2) when $W$ is real, we can often route negative-weighted edges along positive-weighted edges such that the Cheeg…
Most research regarding quantum adiabatic optimization has focused on stoquastic Hamiltonians, whose ground states can be expressed with only real, nonnegative amplitudes. This raises the question of whether classical Monte Carlo algorithms can efficiently simulate quantum adiabatic optimization with stoquastic Hamiltonians. Recent results have given counterexamples in which path integral and diffusion Monte Carlo fail to do so. However, most adiabatic optimization algorithms, such as for solving MAX-k-SAT problems, use k-local Hamiltonians, whereas our previous counterexample for diffusion Mo…
A general time-dependent quantum system can be driven fast from its initial ground state to its final ground state without generating transitions by adding a steering term to the Hamiltonian. We show how this technique can be modified to improve on the standard quantum adiabatic algorithm by making a single-particle and cluster approximation to the steering term. The method is applied to a one-dimensional Ising model in a random field. For the limit of strong disorder, the correction terms significantly enhance the probability for the whole system to remain in the ground state for the proposed…
We compare the critical behavior of the ground state and the thermal state of the XX model. We analyze the full energy spectrum and the eigenstates to reconstruct the ground state and the thermally excited state. With the solutions, we discuss about several physical properties of the states, which are related to quantum phase transition, in various limits, at zero temperature as well as at a thermal equilibrium.
In this paper we prove that a fundamental constraint on the cooling dynamic implies that it is impossible to cool, via a unitary system-bath quantum evolution, a system that is embedded in a thermal environment down to its ground state, if the initial state is a factorized product of system and bath states. The latter is a crucial but artificial assumption often included in many descriptions of system-bath dynamics. The analogous conclusion holds for 'cooling' to any pure state of the system.
Simple and efficient end-to-end quantum thermal and ground state preparation
University of Michigan（密歇根大学）；California Institute of Technology（加州理工学院）；AWS Center for Quantum Computing（亚马逊量子计算中心）；University of California, Berkeley（加利福尼亚大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We study the robustness of quantum information stored in the degenerate ground space of a local, frustration-free Hamiltonian with commuting terms on a 2D spin lattice. On one hand, a macroscopic energy barrier separating the distinct ground states under local transformations would protect the information from thermal fluctuations. On the other hand, local topological order would shield the ground space from static perturbations. Here we demonstrate that local topological order implies a constant energy barrier, thus inhibiting thermal stability.
We have studied the symmetric and non-symmetric pairwise ground state and thermal entanglement in three-qubit anisotropic Heisenberg (XXZ) and Ising in a magnetic field models and in the presence of Dzyaloshinskii-Moriya (DM) interaction. We have found that increasing of the DM interaction and magnetic field can enhance and reduce the entanglement of system. We have shown that the non-symmetric pairwise has higher value concurrence and critical temperature (above which the entanglement vanishes) than the symmetric pairwise. For negative anisotropy the non-symmetric entanglement is a monotonic …
We analyze the thermal Casimir-Polder potential experienced by a ground-state molecule in a planar cavity and investigate the prospects for using such a set-up for molecular guiding. The resonant atom-field interaction associated with this non-equilibrium situation manifests itself in oscillating, standing-wave components of the potential. While the respective potential wells are normally too shallow to be useful, they may be amplified by a highly reflecting cavity whose width equals a half-integer multiple of a particular molecular transition frequency. We find that with an ideal choice of mo…
We address the one-dimensional quantum Ising model as an example of system exhibiting criticality and study in some details the discrimination problem for pairs of states corresponding to different values of the coupling constant. We evaluate the error probability for single-copy discrimination, the Chernoff bound for $n$-copy discrimination in the asymptotic limit, and the Chernoff metric for the discrimination of infinitesimally close states. We point out scaling properties of the above quantities, and derive the external field optimizing state discrimination for short chains as well as in t…
A family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove unive…
We critically discuss the common belief that the thermal Casimir-Polder force can always be represented as a Matsubara sum, as it is true for the Casimir force. On the contrary, an exact treatment of the atom-field coupling reveals that for a ground-state atom, terms associated with virtual-photon absorption contribute to the force which are identified as a signature of non-equilibrium dynamics. Even the equilibrium force on a thermalized atom or molecule may be overestimated when using the ground-state polarizability instead of its thermal counterpart.
We study the robustness of multipartite entanglement of the ground state of the one-dimensional spin 1/2 XY model with a transverse magnetic field in the presence of thermal excitations, by investigating a threshold temperature, below which the thermal state is guaranteed to be entangled. We obtain the threshold temperature based on the geometric measure of entanglement of the ground state. The threshold temperature reflects three characteristic lines in the phase diagram of the correlation function. Our approach reveals a region where multipartite entanglement at zero temperature is high but …
We consider a quantum engine driven by repeated weak interactions with a heat bath of identical three-level atoms. This model was first introduced by Scully et al. [Science, 2003], who showed that coherence between the energy-degenerate ground states serves as a thermodynamic resource that allows operation of a thermal cycle with a coherence-dependent thermalisation temperature. We consider a similar engine out of the quasistatic limit and find that the ground-state coherence also determines the rate of thermalisation, therefore increasing the output power and the engine efficiency only when t…
We establish a connection between ground states of local quantum Hamiltonians and thermal states of classical spin systems. For any discrete classical statistical mechanical model in any spatial dimension, we find an associated quantum state such that the reduced density operator behaves as the thermal state of the classical system. We show that all these quantum states are unique ground states of a universal 5-body local quantum Hamiltonian acting on a (polynomially enlarged) system of qubits arranged on a 2D lattice. The only free parameters of the quantum Hamiltonian are coupling strengthes…
QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most notably the ``Local Hamiltonian'' problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems. The first one is ``Consistency of Local Density Matrices'': given several density matrices describing different (constant-size) subsets of an n-qubit system, decide whether these are consistent with a single global state. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. This uses alg…
We propose a scheme to cool down a mechanical resonator to its quantum ground-state, which is interacting with a working fluid via an optomechanical-like coupling. As opposed to standard laser cooling schemes where coherence renders the motion of a resonator to its ground-state, we consider an incoherent thermal source to achieve the same aim. We show that simultaneous cooling of two degenerate or near-degenerate mechanical resonators is possible, which is otherwise a challenging goal to achieve. The generalization of this method to the simultaneous cooling of multiple resonators is straightfo…
Exact analytical diagonalization is used to study the bipartite entanglement of the antiferromagnetic mixed spin-(1/2,$S$) Heisenberg dimer (MSHD) with the help of negativity. Under the assumption of uniaxial single-ion anisotropy affecting higher spin-$S$ ($S\!>\!1/2$) entities only, the ground-state degeneracy $2S$ is partially lifted and the ground state is two-fold degenerate with the total magnetization per dimer $\pm(S\!-\!1/2)$. It is shown that the largest quantum entanglement is reached for the antiferromagnetic ground state of MSHD with arbitrary half-odd-integer spins $S$, regardles…
We investigate the entanglement properties of a one dimensional chain of spin qubits coupled via nearest neighbor interactions. The entanglement measure used is the n-concurrence, which is distinct from other measures on spin chains such as bipartite entanglement in that it can quantify "global" entanglement across the spin chain. Specifically, it computes the overlap of a quantum state with its time-reversed state. As such this measure is well suited to study ground states of spin chain Hamiltonians that are intrinsically time reversal symmetric. We study the robustness of n-concurrence of gr…
The entanglement in a Hubbard chain of hardcore bosons is investigated. The analytic expression of the global entanglement in ground state is derived. The divergence of the derivative of the global entanglement shows the quantum criticality of the ground state. For the thermal equilibrium state, the bipartite and the multipartite entanglement are evaluated. The entanglement decreases to zero at a certain temperature. The thermal entanglement is rapidly decreasing with the increase of the number of sites in the lattice. The bipartite thermal entanglement approaches a constant value at a certain…
Measurement-based cooling is a method by which a quantum system, initially in a thermal state, can be prepared in its ground state through some sort of measurement. This is done by making a measurement that heralds the system being in the desired state. Here we demonstrate the application of a measurement-based cooling technique to a trapped atomic ion. The ion is pre-cooled by Doppler laser cooling to a thermal state with a mean excitation of $\bar n \approx 18$ and the measurement-based cooling technique selects those occasions when the ion happens to be in the motional ground state. The fid…
We study the entanglement of thermal and ground states in Heisernberg $XX$ qubit rings with a magnetic field. A general result is found that for even-number rings pairwise entanglement between nearest-neighbor qubits is independent on both the sign of exchange interaction constants and the sign of magnetic fields. As an example we study the entanglement in the four-qubit model and find that the ground state of this model without magnetic fields is shown to be a four-body maximally entangled state measured by the $N$-tangle.
The Euclidean formulation of SU(2) Yang-Mills thermodynamics admits periodic, (anti)selfdual solutions to the fundamental, classical equation of motion which possess one unit of topological charge: (anti)calorons. A spatial coarse graining over the central region in a pair of such localised field configurations with trivial holonomy generates an inert adjoint scalar field $ϕ$, effectively describing the pure quantum part of the thermal ground state in the induced quantum field theory. Here we show for the limit of zero holonomy how (anti)calorons associate a temperature independent electric pe…
The question whether a given quantum state is a ground or thermal state of a few-body Hamiltonian can be used to characterize the complexity of the state and is important for possible experimental implementations. We provide methods to characterize the states generated by two- and, more generally, k-body Hamiltonians as well as the convex hull of these sets. This leads to new insights into the question which states are uniquely determined by their marginals and to a generalization of the concept of entanglement. Finally, certification methods for quantum simulation can be derived.
We generalize the formalism proposed by Dalibard, Dupont-Roc, and Cohen-Tannoudji [the DDC formalism] in the fourth order for two atoms in interaction with scalar fields in vacuum to a thermal bath at finite temperature $T$, and then calculate the interatomic interaction energy of two ground-state atoms separately in terms of the contributions of thermal fluctuations and the radiation reaction of the atoms and analyze in detail the thermal corrections to the van der Waals and Casimir-Polder interactions. We discover a particular region, i.e., $\sqrt[4]{λ^3β}\ll L\ll λ$ with $L$, $β$ and $λ$ de…
The goal of adiabatic ground-state preparation is to start a simple quantum system in its ground state and adiabatically evolve the Hamiltonian to a complex one, maintaining the ground state throughout the evolution. In ion-trap-based quantum simulations, coherence times are too short to allow for adiabatic evolution for large chains, so the system evolves diabatically, creating excitations to higher energy states. Because the probability for diabatic excitation depends exponentially on the excitation energy and because the thermal distribution also depends exponentially on the excitation ener…
Dynamical, thermal, and ground-state multiparameter quantum metrology: Fundamental limits and their attainability in magnetometry
Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Università di Bologna（博洛尼亚大学）；BITS Pilani KK Birla Goa Campus（比拉印度理工学院果阿校区）；Harish-Chandra Research Institute（哈里什-钱德拉研究所）；Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences（奥地利科学院维也纳量子光学与量子信息研究所）
