Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians

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 transverse Ising models (TIM) with long-range algebraically decaying interactions as well as disordered XY spin chains with nearest-neighbor interactions. By bounding the convergence time to the equilibrium distribution we rigorously justify the use of PIMC to approximate partition functions and expectations of observables for these models at inverse temperatures that scale at most logarithmically with the number of qubits. The mixing time analysis is based on the canonical paths method applied to the single-site Metropolis Markov chain for the Gibbs distribution of 2D classical spin models with couplings related to the interactions in the quantum Hamiltonian. Since the system has strongly nonisotropic couplings that grow with system size, it does not fall into the known cases where 2D classical spin models are known to mix rapidly.Comment: 26 pages, 2 figures, version published in Quantu

Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy

The relative entropy of a quantum density matrix to a subalgebraic restriction appears throughout quantum information. For subalgebra restrictions given by commuting conditional expectations in tracial settings, strong subadditivity shows that the sum of relative entropies to each is at least as large as the relative entropy to the intersection subalgebra. When conditional expectations do not commute, an inequality known as quasi-factorization or approximate tensorization replaces strong subadditivity. Multiplicative or strong quasi-factorization yields relative entropy decay estimates known as modified logarithmic-Sobolev inequalities for complicated quantum Markov semigroups from those of simpler constituents. In this work, we show multiplicative comparisons between subalgebra-relative entropy and its perturbation by a quantum channel with corresponding fixed point subalgebra. Following, we obtain a strong quasi-factorization inequality with constant scaling logarithmically in subalgebra index. For conditional expectations that nearly commute and are not too close to a set with larger intersection algebra, the shown quasi-factorization is asymptotically tight in that the constant approaches one. We apply quasi-factorization to uncertainty relations between incompatible bases and to conditional expectations arising from graphs.Comment: 29 pages, 1 figure; updated to reflect recent developments and resulting improvement

Spectral estimation for stoquastic Hamiltonians: a comparison between classical imaginary-time evolution and quantum real-time evolution

We present a classical Monte Carlo (MC) scheme which efficiently estimates an imaginary-time, decaying signal for stoquastic (i.e. sign-problem-free) local Hamiltonians. The decay rates in this signal correspond to Hamiltonian eigenvalues (with associated eigenstates present in an initial state) and can be classically extracted using a classical signal processing method like ESPRIT. We compare the efficiency of this MC scheme to its quantum counterpart in which one extracts eigenvalues of a general local Hamiltonian from a real-time, oscillatory signal obtained through quantum phase estimation circuits, again using the ESPRIT method. We prove that the ESPRIT method can resolve S = poly(n) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) quantum and classical effort through the quantum phase estimation circuits. We prove that our Monte Carlo scheme plus the ESPRIT method can resolve S = O(1) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) purely classical effort for stoquastic Hamiltonians. These results thus quantify some opportunities and limitations of classical Monte Carlo methods for spectral estimation of stoquastic Hamiltonians. We numerically compare these MC and QPE eigenvalue estimation schemes by implementing them for an archetypal stoquastic Hamiltonian system: the transverse field Ising chain

Spectral estimation for Hamiltonians: A comparison between classical imaginary-time evolution and quantum real-time evolution

We consider the task of spectral estimation of local quantum Hamiltonians. The spectral estimation is performed by estimating the oscillation frequencies or decay rates of signals representing the time evolution of states. We present a classical Monte Carlo (MC) scheme which efficiently estimates an imaginary-time, decaying signal for stoquastic (i.e. sign-problem-free) local Hamiltonians. The decay rates in this signal correspond to Hamiltonian eigenvalues (with associated eigenstates present in an input state) and can be extracted using a classical signal processing method like ESPRIT. We compare the efficiency of this MC scheme to its quantum counterpart in which one extracts eigenvalues of a general local Hamiltonian from a real-time, oscillatory signal obtained through quantum phase estimation circuits, again using the ESPRIT method. We prove that the ESPRIT method can resolve S = poly(n) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) quantum and classical effort through the quantum phase estimation circuits, assuming efficient preparation of the input state. We prove that our Monte Carlo scheme plus the ESPRIT method can resolve S = O(1) eigenvalues, assuming a 1/poly(n) gap between them, with poly(n) purely classical effort for stoquastic Hamiltonians, requiring some access structure to the input state. However, we also show that under these assumptions, i.e. S = O(1) eigenvalues, assuming a 1/poly(n) gap between them and some access structure to the input state, one can achieve this with poly(n) purely classical effort for general local Hamiltonians. These results thus quantify some opportunities and limitations of Monte Carlo methods for spectral estimation of Hamiltonians. We numerically compare the MC eigenvalue estimation scheme (for stoquastic Hamiltonians) and the quantum-phase-estimation-based eigenvalue estimation scheme by implementing them for an archetypal stoquastic Hamiltonian system: the transverse field Ising chain

Classical simulations of quantum systems using stabilizer decompositions

One of the state of the art techniques for classically simulating quantum circuits relies on approximating the output state of the circuit by a superposition of stabilizer states. If the number of non-Clifford gates in the circuit is small, such simulations can be very effective. This thesis provides various improvements in this framework. First, we describe an improved method of computing approximate stabilizer decompositions, which reduces the time cost of computing a single term in the decomposition from $O(\ell n^2)$ to $O(m n^2)$, where $\ell$ is the total number of gates in the circuit, and $m$ is the number of non-Clifford gates. Since this subroutine has to be repeated exponentially many times, this improvement can be significant in practice whenever $\ell \gg m$. Our method uses a certain re-writing of the circuit, which in some cases allows for a significant amelioration of the exponential scaling of the required classical resources. Furthermore, we describe a method of constructing exact, low-rank stabilizer decompositions of $\ket{\psi}^{\otimes m}$, where $\ket{\psi}$ is either a magic state or an equatorial state. For any single qubit magic state $\ket{\psi}$, we find stabilizer decompositions of $\ket{\psi}^{\otimes m}$ with $2^{m\log_2(3)/4}$ terms. This improves on the best known bound of $2^{m \log_2(7)/6}$. Similarly, for any single qubit equatorial state $\ket{\psi}$, we give a stabilizer decomposition of $\ket{\psi}^{\otimes m}$ with $2^{m/2}$ terms. To our knowledge no such decompositions were previously known. These results translate to milder exponential scaling of the classical resources required for estimating probabilities of quantum circuits up to a polynomially small multiplicative error, as well as allowing more types of circuits to be simulated in this way. We also consider certain obstructions to classical simulations. It has been argued in various contexts that contextuality and non-locality hamper classical simulations of quantum circuits. Linear constraint systems (LCSs) are a generalization of the well-known Peres-Mermin magic square, which has been recently used to prove a separation between the power of constant depth classical and quantum circuits. While binary LCSs have been studied in detail, $d$-ary LCSs are less well-understood. In this thesis we consider linear constraint systems modulo $d > 2$. We give a simple proof, of the previously known fact, that any linear constraint system which admits a quantum solution consisting of generalized Pauli observables in odd dimension must be classically satisfiable. We further prove that, for odd $d$, if a Pauli-like commutation relation between two variables in the LCS arises, then it has no quantum solutions in any dimensions, in stark contrast to the even $d$ case. We apply this result to various examples, for instance showing that many generalizations of the Peres-Mermin magic square do not give rise to a quantum vs. classical satisfiability gap