Why and When Is Pausing Beneficial in Quantum Annealing?
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Why and when pausing is beneficial in quantum annealing Huo Chen1, 2 and Daniel A. Lidar1, 2, 3, 4 1Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA 2Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA 3Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA 4Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA Recent empirical results using quantum annealing hardware have shown that mid anneal pausing has a surprisingly beneficial impact on the probability of finding the ground state for of a variety of problems. A theoretical explanation of this phenomenon has thus far been lacking. Here we provide an analysis of pausing using a master equation framework, and derive conditions for the strategy to result in a success probability enhancement. The conditions, which we identify through numerical simulations and then prove to be sufficient, require that relative to the pause duration the relaxation rate is large and decreasing right after crossing the minimum gap, small and decreasing at the end of the anneal, and is also cumulatively small over this interval, in the sense that the system does not thermally equilibrate. This establishes that the observed success probability enhancement can be attributed to incomplete quantum relaxation, i.e., is a form of beneficial non-equilibrium coupling to the environment. I. INTRODUCTION lems [23] and training deep generative machine learn- ing models [24]. Numerical studies [25, 26] of the p-spin Quantum annealing [1{4] stands out among the multi- model also agree with these empirical results. However, tude of concurrent approaches being developed to explore despite a useful qualitative explanation offered for the quantum computing, as having achieved the largest scale thermalization mechanism by which pausing improves to date when measured in terms of the sheer number success probabilities [21], a thorough analysis of the ex- of controllable qubits. Today's commercial quantum an- act mechanism of this phenomenon is still lacking. Here nealers feature thousands of superconducting flux qubits we provide such an analysis, and identify sufficient con- and are being used routinely to test whether this ap- ditions for pausing to provide an enhancement. proach can provide a quantum advantage over classical Our analysis is based on a detailed investigation of a computing [5{11]. While there is no consensus that such quantum two-level system model coupled to an Ohmic an advantage has been demonstrated, there is significant bath. The two-level system can be either a single qubit progress on the development of \software" methods that or a multi-qubit system whose lowest two energy levels improve quantum annealing performance. Such meth- are separated by a large gap from the rest of the spec- ods take advantage of the advanced control capabilities trum. The analysis builds on tools from the theory of of quantum annealers to implement protocols that result open quantum systems [27{29], specifically master equa- in higher success probabilities, shorter times to solution, tions appropriate for time-dependent (driven) Hamiltoni- faster equilibration, etc. Continued progress in this di- ans [30{35]. Through numerical investigation we identify rection is clearly critical as a complementary approach to a set of sufficient conditions, stated in term of the prop- improving the underlying hardware by reducing physical erties of the relaxation rate along the anneal, and prove source of noise and decoherence. a theorem guaranteeing that it is advantageous to pause Among the various empirical protocols that have been mid-anneal. The advantage gained is a higher success developed to improve the performance of quantum an- probability than is attainable without pausing. nealing, such as error suppression and correction [12{15] We thus establish, in a rigorous sense, that there ex- arXiv:2005.01888v2 [quant-ph] 5 Aug 2020 and inhomogeneous driving [16{19], the mid-anneal paus- ists a non-trivial optimal pausing point under a set of ing protocol stands out as particularly powerful. Pausing reasonable assumptions. We do not identify the optimal superficially resembles the idea of slowing down near the pausing duration, but we do prove that the optimal paus- minimum gap, as in the optimal schedule for the Grover ing point occurs after the minimum gap, in accordance problem [3, 20], but the context here is entirely different with the prior empirical and numerical evidence. This due to the fact that pausing happens in an open sys- result is stated in Theorem1 below, which can be sum- tem subject to thermal relaxation. The first study [21] marized as saying that an optimal pausing point exists if to systematically test this approach empirically using a the relaxation rate right after the minimum gap is large D-Wave 2000Q device [22], demonstrated a dramatic im- relative to the pause duration but small at the end of the provement in the probability of finding the ground state anneal, decreases right after crossing the minimum gap (i.e., the success probability) when an anneal pause was and also at the end of the anneal, and is also cumulatively inserted shortly after crossing the minimum gap. Follow- small over this interval, in the sense that the system does up studies confirmed that pausing is advantageous on not fully thermally equilibrate. different problems such as portfolio optimization prob- The structure of this paper is as follows. In Sec.II we 2 define our model of the two-level system. In the multi- where Ω(s) and θ(s) are a reparameterization of the qubit case this involves deriving an effective Hamiltonian annealing schedules: A(s) = Ω(s) cos θ(s) and B(s) = for the projection to the low energy subspace of the full Ω(s) sin θ(s). (see AppendixA for a detailed explana- Hamiltonian. We introduce a certain parametrization of tion). We call θ(s) the annealing angle and θ_(s) the an- the gap and the geometric phase that ensures the problem gular progression. The term θY=_ 2 has its origin as a geo- is hard for quantum annealing, in the sense that the suc- metric phase [37]. Loosely, Ω(s) corresponds to the time- cess probability is low even on a timescale that is large dependent gap and dθ=ds corresponds to how fast/slow compared to the inverse of the minimum spectral gap the Hamiltonian changes. For a typical single qubit an- along the anneal. In Sec. III we treat the same model nealing process, the boundary condition θ(0) = 0 and as an open quantum system using master equation tech- θ(1) = π=2 needs to be satisfied (noticing that in Eq. (1) niques, specifically the Redfield equation with and with- we permuted the Pauli X and Z matrices in the standard out the rotating wave approximation, and the adiabatic notation). master equation. Then, in Sec.IV we introduce a pause into the annealing schedule and study its effects. We first demonstrate numerically that an optimal pausing posi- B. Projected TLS from a multi-qubit model tion exists before the end of of the anneal, depending on the monotonicity properties of the relaxation rate after For general multi-qubit annealing, the Hamiltonian is the minimum gap is crossed. Building on these obser- vations we then prove a theorem establishing sufficient X H (s) = A(s)H + B(s)H = E (s) jn(s)ihn(s)j : conditions for the existence of such an optimal pausing S d p n n point. We conclude in Sec.V, and present additional (3) technical details in the Appendix. where fjn(s)ig is the instantaneous energy eigenbasis and En(s) are the instantaneous energies. Hd and Hp are the driver and problem Hamiltonian, respectively. Hence- T II. \HARD" SINGLE-QUBIT AND forth we assume that HS = (HS) , i.e., that HS(s) is MULTI-QUBIT CLOSED SYSTEM MODELS real for all s. The system density matrix can be written in the instantaneous energy eigenbasis: In this section we consider two scenarios: single qubit X annealing, and a projected two-level system (TLS) aris- ρ(s) = ρnm jnihmj : (4) ing from multi-qubit annealing. We define a model that nm makes these problems \hard" for quantum annealing, in the sense of a small success probability even over a We call the associated matrixρ ~ = [ρnm] the density ma- trix in the adiabatic frame, and show in AppendixB that timescale that is long compared to the heuristic adiabatic h i timescale (given by the inverse of the minimum gap along it obeys the von Neumann equation ρ~_ = −i H;~ ρ~ with the anneal path). the effective Hamiltonian 0 1 tf E0 −i 0 1_ ::: _ A. Single qubit H~ = Bi 0 1 tf E1 :::C (5) @ . A . .. We write the single qubit annealing Hamiltonian in the form If we truncate the effective Hamiltonian (5) to the lowest two energy levels and shift it by a constant term, we find: 1 HS(s) = − A(s)Z + B(s)X ; (1) 2 ~ tf Ω(s) H2 = 0 1_ Y − Z; (6) 2 where s = t=tf is the dimensionless instantaneous time, t is the actual time, tf is the total anneal time, and where Ω(s) = E1 − E0 is the energy gap between the A(s) and B(s) are the annealing schedules. Note that lowest two energy levels. We call this the projected TLS we permuted the Pauli matrices X and Z of the con- Hamiltonian. An alternative way to derive this effective ventional single qubit annealing Hamiltonian in order to Hamiltonian is via the well-known adiabatic intertwiner have the same expression for both the single qubit and (see, e.g., Ref.