An energetic perspective on rapid quenches in quantum annealing
Adam Callison, Max Festenstein, Jie Chen, Laurentiu Nita, Viv Kendon, Nicholas Chancellor
AAn energetic perspective on rapid quenches inquantum annealing
Adam Callison , Max Festenstein , , Jie Chen , LaurentiuNita , Viv Kendon , and Nicholas Chancellor Blackett Laboratory, Imperial College London, London SW7 2BW, UK Department of Physics; Joint Quantum Centre (JQC) Durham-Newcastle,Durham University, South Road, Durham, DH1 3LE, UKE-mail: [email protected] , [email protected] Abstract.
There are well developed theoretical tools to analyse how quantumdynamics can solve computational problems by varying Hamiltonian parametersslowly, near the adiabatic limit. On the other hand, there are relatively few toolsto understand the opposite limit of rapid quenches, as used in quantum annealingand (in the limit of infinitely rapid quenches) in quantum walks. In this paper,we develop several tools which are applicable in the rapid quench regime. Firstly,we analyse the energy expectation value of different elements of the Hamiltonian.From this, we show that monotonic quenches, where the strength of the problemHamiltonian is consistently increased relative to fluctuation (driver) terms, willyield a better result on average than random guessing. Secondly, we developmethods to determine whether dynamics will occur locally under rapid quenchHamiltonians, and identify cases where a rapid quench will lead to a substantiallyimproved solution. In particular, we find that a technique we refer to as “pre-annealing” can significantly improve the performance of quantum walks. We alsoshow how these tools can provide efficient heuristic estimates for Hamiltonianparameters, a key requirement for practical application of quantum annealing. a r X i v : . [ qu a n t - ph ] A ug n energetic perspective on rapid quenches in quantum annealing Contents1 Introduction 22 Rapid quench examples 5
A.1 Energy conservation mechanism . . . . . . . . . . . . . . . . . . . . . . 24A.2 Energy redistribution mechanism . . . . . . . . . . . . . . . . . . . . . 25A.3 The case of B ( t ) →
0: divergence of Γ . . . . . . . . . . . . . . . . . . 26
Appendix B Lower bound on the average dynamic coefficient 27
B.1 Bound on probabilities in a range based on second moment . . . . . . 27B.2 A simple lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1. Introduction
Quantum computing using continuous time evolution has gained much interest inrecent years. This includes adiabatic quantum computing (Farhi et al., 2000), quantumannealing (Finnila et al., 1994; Kadowaki and Nishimori, 1998), and continuous-timequantum walks (Farhi and Gutmann, 1998). Applications of quantum annealing havebeen explored in many diverse fields including traditional computer science (Chancelloret al., 2016a; Choi, 2010, 2011), decoding communications (Chancellor et al., 2016b),finance (Marzec, 2016; Or´us et al., 2019; Venturelli and Kondratyev, 2019), errorcorrection of quantum memories (Roffe et al., 2019), scheduling (Venturelli et al., 2000;Crispin and Syrichas, 2013; Tran et al., 2016), computational biology (Perdomo-Ortizet al., 2012), flight gate assignment (Stollenwerk et al., 2019), air traffic management(Stollenwerk et al., 2020), and hydrology (OMalley, 2018). This has partially beenbecause of the success of experimental quantum annealing, a notable example being n energetic perspective on rapid quenches in quantum annealing problem
Hamiltonian H prob , which is diagonal in the computational basis, andencodes the classical problem being solved, and a driver Hamiltonian H drive whichimplements quantum dynamics to explore the solution space. We use two equivalentforms for the total Hamiltonian. First, H AB ( t ) = A ( t ) H drive + B ( t ) H prob , (1)where A ( t ) and B ( t ) are positive, time-dependent control functions. However,typically the crucial feature is what happens to the ratio of driver to problem strength A ( t ) / B ( t ) as the algorithm progresses. As such, we define an alternative parametrizationof the Hamiltonian, up to an overall (time-dependent) scaling factor B ( t ), as H Γ ( t ) = Γ( t ) H drive + H prob , (2)where there is a single control function Γ( t ) > A ( t ) / B ( t ) . Since (1) and(2) are equivalent, up to a rescaling of the time parameter, results for one form ofHamiltonian will generalize to results for the other. We use both forms, choosing themost convenient for the specific problem or example.Hamiltonians of the form (1) and (2), which begin with A ( t ) > B ( t ) = 0and end with A ( t ) = 0 and B ( t ) >
0, or equivalently, begin with Γ( t ) (cid:29) t ) = 0, are used for most types of continuous-time quantum computing. Whenrun on a much shorter timescale than required for adiabatic quantum computing, wecall this a rapid quench . n energetic perspective on rapid quenches in quantum annealing continuous time quantum walk (QW) introduced by Farhi and Gutmann (1998);Childs and Goldstone (2004), in which the control functions are time-independentand set so that Γ( t ) = γ where γ is a constant hopping rate. This can be viewed asthe limit of an infinitely fast quench, in which B (0) jumps from zero to A (0) /γ at t = 0and A ( t f ) drops to zero at the final time t f . The other pure state continuous-timequantum computing which is commonly considered is adiabatic quantum computing (AQC) introduced by Farhi et al. (2000), for which the control functions A ( t ) and B ( t ) are varied slowly from A (0) = 1 and B (0) = 0 to A ( t f ) = 0 and B ( t f ) = 1.By the adiabatic theorem of quantum mechanics, this achieves a success probability(probability of finding the ground state of the problem Hamiltonian H prob ) whichapproaches 1 as t f → ∞ . For a review of AQC see Albash and Lidar (2018). For athorough discussion of the relationship between AQC and QW, see the introductionsof Morley et al. (2019); Callison et al. (2019). The fully coherent regime has provablequantum speedups in the case of both AQC and QW. For instance, unstructuredsearch, the continuous time analog of Grover’s search, can yield the same speedup inthe AQC (Roland and Cerf, 2002) and QW (Childs and Goldstone, 2004) settings asthe gate based counterpart. It is possible to interpolate between these two techniqueswhile preserving the speedup (Morley et al., 2019).For problems which are closer to real world optimisation, theoretical studies havemostly focused on AQC (Albash and Lidar, 2018), likely because the adiabatic theoremprovides a general way to show that such algorithms could in principle succeed withhigh probability. While theoretically tractable, the adiabatic regime is difficult toreach experimentally, and contains some counter-intuitive effects in the deep adiabaticregime (Wiebe and Babcock, 2012; Campos Venuti and Lidar, 2018; Passos et al.,2020). Solving NP-hard problems adiabatically will at most obtain a polynomialspeed up (assuming P (cid:54) = NP). Since AQC requires the system to remain coherentthroughout, exponentially long runtime requires exponentially long coherence time,which is experimentally challenging for near-term quantum computing. When theruntime is limited by a constant or mildly scaling coherence time, such an algorithmcould only solve the problem with an exponentially low probability, and thereforerequire exponentially many repeats to succeed with high probability. This approach,however, is a valid one for problems other than search. Recent numerical results onspin-glasses using QW show favourable scaling from many short run repeats (Callisonet al., 2019). It has also been numerically demonstrated that rapid quenches can besuperior to long quenches for AQC-like algorithms (Crosson et al., 2014). Finally,for single shot, high success probability algorithms for NP-hard problems, achievingeven a polynomial speedup typically requires setting, with exponential precision, thecontrol functions to values which lead to exponential small gaps in the Hamiltonianspectrum. This was shown to be necessary for unstructured search in Childs et al.(2002); Roland and Cerf (2002); Morley et al. (2019) and for the random energy model(Farhi et al., 2008) in Callison et al. (2019). This requirement is problematic, as thereare no general methods for determining where these gaps occur and because suchprecise control settings can be difficult to achieve in real hardware. Recent work byChakraborty et al. (2018) demonstrates that some of the fine tuning requirements inunstructured search can be avoided by formulating the Hamiltonian differently, it isunclear whether this approach would extend to the random energy model of Farhiet al. (2008).Given the near term importance of methods which can succeed with limited n energetic perspective on rapid quenches in quantum annealing
2. Rapid quench examples
To motivate our theoretical tools, we start with three illustrative examples showing thepower of rapid quenches to solve problems. For simplicity and concreteness, we focuson monotonic quenches; that is, quenches for which the control parameter Γ( t (cid:48) ) ≤ Γ( t ) ∀ t (cid:48) > t . This is a minimal modification to the time-independent continuous time quantumwalk. It consists of two time-independent stages of evolution separated by an infinitelyfast quench. Because each stage is effectively a continuous time quantum walk, werefer to this as a two-stage quantum walk. We use a simple transverse field driverHamiltonian H drive = n − n (cid:88) j =1 ˆ X j , (3) n energetic perspective on rapid quenches in quantum annealing t Γ Figure 1.
The annealing schedule for the two stage quantum walks in Fig. 2(red, solid lines), Fig. 3 (blue, dot dashed lines), and Fig. 4 (magenta, dashedlines). In all cases the step occurs at t = 10 (dotted line). where is the identity operator and ˆ X j is the Pauli ˆ X operator acting in qubit j , but,instead of using a constant control function (Γ( t ) = γ ), we use the time-dependentschedule Γ( t ) = (cid:40) γ < t < t γ t < t < ( t + t ) , (4)which consists of two consecutive evolution stages with two different time independentHamiltonians. Each of these stages is effectively a quantum walk, although the secondstage uses non-standard starting conditions as its initial state is the final state of thefirst stage. The standard initial state is the equal superposition of all basis states, | ψ (cid:105) = 2 − n/ (cid:80) j | j (cid:105) , chosen because it is the ground state of the driver Hamiltonian,and also represents our ignorance of which basis state is the solution to the problem.The schedules we use for the two stage quantum walks are shown in Fig. 1 for each ofour three examples.As discussed in Callison et al. (2019) a quantum walk can be understood froman energetic perspective according to a mechanism referred to there as the energyconservation mechanism . Being time-independent, quantum walks conserve the totalenergy of the system. To show the effect of changing the hopping rate γ part waythrough the walk, thus disrupting the energy conservation, our first example is a simpletwo qubit problem Hamiltonian H (2 Q )prob = − ˆ Z ˆ Z −
12 ˆ Z , (5)where ˆ Z j is the Pauli ˆ Z operator acting on qubit j . We start the system at t = 0in the state | ψ (cid:105) = ( | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) ), the two qubit ground state of thedriver Hamiltonian H drive in (3). To simplify notation, we define (cid:104) H prob (cid:105) ψ ( t ) ≡(cid:104) ψ ( t ) | H prob | ψ ( t ) (cid:105) , the instantaneous expectation value of the problem Hamiltonianwith respect to the state ψ ( t ) at time t . Likewise, (cid:104) H drive (cid:105) ψ ( t ) ≡ (cid:104) ψ ( t ) | H drive | ψ ( t ) (cid:105) for the driver Hamiltonian. We have the total energy E Γ ( t ) = Γ( t ) (cid:104) H drive (cid:105) + (cid:104) H prob (cid:105) .Fig. 2 (top) shows that the expectation value (cid:104) H drive (cid:105) for the transverse field iszero initially ( t = 0). As in Callison et al. (2019), the energy conservation mechanism n energetic perspective on rapid quenches in quantum annealing E n e r g y e x p ec t a t i o n t P ( t ) Figure 2.
Two stage quantum walk using Hamiltonian in (5) with γ = 2and γ = . The instantaneous quench occurs at time t = 10 (vertical dottedline). Top: energy expectation values E Γ = Γ (cid:104) H drive (cid:105) + (cid:104) H prob (cid:105) (gold), Γ (cid:104) H drive (cid:105) (green), (cid:104) H prob (cid:105) (blue). Also shown (black, dashed) is a guide to the eye at 0,and the minimum eigenvalue of H prob (red, dashed). Bottom: probability P ( t )of being in the ground state of H prob at time t (blue), probability of randomguessing (red, dashed). then decreases the expectation value of the problem Hamiltonian at the expense ofincreasing the expectation value of the driver Hamiltonian. When the instantaneousquench is performed, the problem Hamiltonian expectation value is unchanged, butthe driver Hamiltonian expectation value (and therefore the total energy expectationvalue E Γ ( t )) is reduced. As the minimum eigenvalue of H drive is zero, the total energyexpectation value E Γ ( t ) acts as an effective upper bound on (cid:104) H prob (cid:105) ψ ( t ) . The neteffect is that, even if all of the energy stored in the transverse field were returnedto the problem Hamiltonian, its expectation value would still be less than it wasat the beginning of the algorithm. What actually happens, however, is that thetransverse field is able to capture even more of the energy, thereby reducing theproblem Hamiltonian expectation value further, and increasing the average probabilityof finding the ground state, Fig.2 (bottom).A more realistic problem is the Sherrington-Kirkpatrick spin-glass (Sherringtonand Kirkpatrick, 1975) ground-state problem investigated in Callison et al. (2019).This has the problem Hamiltonian H ( SK )prob = − n − (cid:88) ( a (cid:54) = b )=0 J ab ˆ Z a ˆ Z b − n − (cid:88) b =0 h b ˆ Z b , (6)the couplings J ab and fields h b are drawn independently from the normal distribution N (0 , σ ) with mean 0 and variance σ .Figure 3 shows a two stage quantum walk performed on a nine qubit Sherrington-Kirkpatrick Hamiltonian ‡ from the public repository in Chancellor et al. (2019) whichis associated with Callison et al. (2019). In the setting of this larger problem,the fluctuations after each stage of the quantum walk are smaller relative to thedynamical range than in the two qubit case, a very early sign of the approach to thethermodynamic limit. Apart from this, the behaviour is qualitatively similar to the ‡ We chose instance ovcjhwbhtcpcvwicoxpdpvjzqojril and used it throughout this paper for allsingle SK problem examples. n energetic perspective on rapid quenches in quantum annealing E n e r g y e x p ec t a t i o n t P ( t ) Figure 3.
Two stage quantum walk on a 9 qubit Sherrington-Kirkpatrick spinglass, ID code ovcjhwbhtcpcvwicoxpdpvjzqojril from the public repositoryin Chancellor et al. (2019) with γ = 4 and γ = 1. The instantaneousquench occurs at time t = 10, (vertical dotted line). Top: energy expectations E Γ = Γ (cid:104) H drive (cid:105) ψ ( t ) + (cid:104) H prob (cid:105) ψ ( t ) (gold), Γ (cid:104) H drive (cid:105) ψ ( t ) (green), (cid:104) H prob (cid:105) ψ ( t ) (blue). Also shown (black, dashed) is a guide to the eye at 0, and the minimumeigenvalue of H prob (red, dashed). Bottom: probability P ( t ) of being in theground state of H prob at time t (blue). two qubit toy model H (2 Q )prob of (5) shown in Fig. 2, and produces a significant increasein the probability of finding the ground state. We introduce is a biased driver Hamiltonian, similar to the one used in Duan et al.(2013); Graß (2019). We formulate our biased driver Hamiltonian slightly differentlyas H bias ( g, θ ) = n − n (cid:88) i =1 (cid:16) cos( θ ) ˆ X i + g i sin( θ ) ˆ Z i (cid:17) , (7)where g i ∈ {− , } is a candidate (or guess) solution, and takes the value 1 if the i th bit of the guess solution is 0, and − ≤ θ ≤ π ; if θ = 0 the guess goes unused and the driver reducesto a transverse field of (3). In the other extreme, if θ = π , then the ground stateof H bias ( g, θ = π ) is the candidate solution and there are no dynamics. The groundstate of the biased driver Hamiltonian has zero energy for all allowed values of θ and g ,and is a tensor product of spin states which are each anti-parallel to the fields in (7);this state is used as the initial state. For simplicity, in this example we only considerbiasing toward the most optimal solution (i.e., correct guesses), and we use the samenine-qubit SK spin glass as in the previous subsection.As Fig. 4 shows, the effect of biasing toward the optimal solution is to lower theinitial values of E Γ and (cid:104) H prob (cid:105) ψ ( t ) ; biasing toward a well chosen guess effectivelygives the algorithm a ‘head start’ with respect to energy expectation values. This isqualitatively similar to what happens at the beginning of the second stage of the twostage quantum walk, except that the driver energy (cid:104) H bias ( g, θ ) (cid:105) ψ ( t ) starts at exactlyzero, rather than having some initial energy left over from a previous stage. The biasimproves the initial stage success probability by a factor of ten compared with the n energetic perspective on rapid quenches in quantum annealing E n e r g y e x p ec t a t i o n t P ( t ) Figure 4.
Biased two stage quantum walk on a 9 qubit Sherrington-Kirkpatrick spin glass, ID code ovcjhwbhtcpcvwicoxpdpvjzqojril from thepublic repository in Chancellor et al. (2019) with γ = 3 and γ = 1, using abiased driver (7), biased towards the optimal solution of H prob using θ = π . Theinstantaneous quench occurs at time t = 10 (vertical dotted line). Top: energyexpectations E walk = Γ (cid:104) H drive (cid:105) ψ ( t ) + (cid:104) H prob (cid:105) ψ ( t ) (gold), (Γ (cid:104) H drive (cid:105) ψ ( t ) (green), (cid:104) H prob (cid:105) ψ ( t ) (blue). Also shown (black, dashed) is a guide to the eye at 0, and theminimum eigenvalue of H prob (red, dashed). Bottom: probability P ( t ) of beingin the ground state of H prob at time t . (blue). unbiased walk in Fig. 3, while the second stage again provides a (further) factor ofthree improvement. This biased two-stage quantum walk example provides proof-of-concept that the mechanism we describe can be leveraged on top of a biased search.A thorough analysis of biased (single stage) quantum walks as a subroutine for hybridquantum/classical computing is forthcoming (Nita et al., 2020). Our final example is again in two stages, but this time the first stage is a quantumanneal, and the second stage is a quantum walk that starts from the point where theanneal stops. The motivating intuition is that the initial time-dependent annealingstage will prepare an initial state for the quantum walk that has a lower averageproblem energy (cid:104) H prob (cid:105) ψ ( t ) than the usual uniform superposition state. If performedtoo slowly, such a quench will put the system into its instantaneous ground state,by the adiabatic theorem of quantum mechanics, and there will be no quantumwalk dynamics. If performed too rapidly, the state will not evolve much during theanneal stage and the resulting quantum walk will be similar to one without a pre-annealing stage. However, if the anneal is performed at an intermediate rate, it leadsto significant quantum walk dynamics, starting from a lower problem Hamiltonianexpectation value (cid:104) H prob (cid:105) ψ ( t ) .Using the H AB parametrization defined in (1), we consider pre-annealing with aquadratic schedule for a time t , and then a steady state quantum walk afterwards;specifically, we define the schedule A ( t ) = (cid:40) γ [1 + ( tt − ] 0 ≤ t ≤ t γ t < t < ( t + t ) , (8) n energetic perspective on rapid quenches in quantum annealing t A ( t ) , B ( t ) Figure 5.
Schedule A ( t ) (solid) and B ( t ) (dashed) of a pre-annealed quantumwalk using γ ≈ .
004 and t = 4 (blue, vertical dotted line), t = 0 . t = 0, (red, pure QW). B ( t ) = (cid:40) [1 − ( tt − ] 0 ≤ t ≤ t t < t < ( t + t ) (9)which is plotted in Fig. 5 for the values of t we use.Using the same nine-qubit SK problem as before, with its optimal γ value ofapproximately 1 . t are shown in Fig. 6.Pre-annealing both decreases the average problem expectation value (cid:104) H prob (cid:105) ψ ( t ) andincreases the success probability, but causes the peak values to be reached moreslowly. In the longest pre-anneal with t = 4, the success probability undergoessmall amplitude, approximately sinusoidal, oscillations suggesting that the dynamicsare dominated by a two level subspace. For t = 0 . t = 0, the oscillations areless structured, indicating that more than two energy levels are playing a non-trivialrole in the dynamics.The increases in the success probability seen in Fig. 6 are relatively modest forthis example. To determine the typical improvement in success probability due topre-annealing, we use all 10 ,
000 Sherrington-Kirkpatrick instances from Chancelloret al. (2019) at each size from n = 5 to n = 11 and compare the quantum walksuccess probability averaged over the quantum walk stage using 20 different linearlyspaced pre-annealing times up to t = 4. In Fig. 7 (top), we see that the successprobability increases with pre-anneal time, up to a plateau, and the relative effect ofpre-annealing becomes larger as n increases. To quantify this effect, we calculate thescaling exponent at each pre-annealing time by fitting a linear model on log-linearaxes. We find a scaling exponent κ such that the success probability p success ∝ κn .The fitted values of κ are plotted in Fig. 7 (bottom). As the inset of Fig. 7 (bottom)shows, the success probability is modelled well by a simple exponential function, as inCallison et al. (2019). We find that pre-annealing significantly improves the scalingfrom κ = − .
418 for a pure quantum walk, in agreement with Callison et al. (2019),to a maximum of κ ≈ − . n energetic perspective on rapid quenches in quantum annealing E n e r g y e x p ec t a t i o n t P ( t ) Figure 6.
Pre-anneal performed on a nine qubit Sherrington-Kirkpatrick spinglass, ID code ovcjhwbhtcpcvwicoxpdpvjzqojril from Chancellor et al. (2019),for pre-anneal times t = 4 (blue), t = 0 . t = 0 (red), i.e., purequantum walk. Dotted lines show when the pre-anneal ends. Top: Expectationvalues E Γ ( t ) = AB (cid:104) H drive (cid:105) ψ ( t ) + (cid:104) H prob (cid:105) ψ ( t ) (dot-dashed), (cid:104) H prob (cid:105) ψ ( t ) (solid), AB (cid:104) H drive (cid:105) ψ ( t ) (dashed, colour). The black dashed line indicates the minimumeigenvalue of H prob . Bottom: success probability P ( t ) to be in the lowesteigenstate of H prob at time t . t -1 › P fi -1 › P fi Figure 7.
Top: success probability (cid:104) P (cid:105) for n = 5 to n = 11 for 21 differentlinearly spaced pre-anneal times from t = 0 to t = 4, darker magenta colourindicates higher n . All data are averaged over all 10 ,
000 Sherrington-Kirkpatrickinstances from Chancellor et al. (2019) at each size. Bottom: Scaling exponent κ for a model where p success ∝ κn extracted from the linear fit on log-linear axesfor different pre-annealing times in the inset. Inset: Scaling of success probabilityversus n , for the same t values, with t = 0 in red, and t = 4 in dark blue (samecolour coding as the bottom main figure). n energetic perspective on rapid quenches in quantum annealing
3. Energy redistribution mechanism
In all the examples in section 2, we observe that the total energy expectation value E Γ ( t ) never increases during a rapid quench, and that E Γ ( t ) serves as an upper-bound to the problem expectation value (cid:104) H prob (cid:105) ψ ( t ) , assuming that the groundstateof H drive is arranged to be at zero energy (the identity term in (3) ensures this). Inthis section, we formalise these observations into a mechanism that we refer to asthe energy redistribution mechanism . Our analysis extends the energy conservationarguments made in Hastings (2019); Callison et al. (2019) to quenches where theHamiltonian is not time invariant, and therefore total energy is not conserved.Consider a closed system quantum annealing schedule on a system with aHamiltonian H ( t ) defined by (2): H ( t ) = Γ( t ) H drive + H prob . We show that (for duration t f ≥
0) the energy expectation value with respect to theproblem Hamiltonian at the end is never higher than at the initial time t = 0, (cid:104) ψ ( t f ) | H prob | ψ ( t f ) (cid:105) ≤ (cid:104) ψ (0) | H prob | ψ (0) (cid:105) , (10)provided the following conditions are satisfied:(i) ( initial ground state ) the initial state | ψ ( t = 0) (cid:105) is a ground state of the driverHamiltonian H drive (ii) ( positivity ) the control function is non-negative: Γ( t ) ≥ ∀ t (iii) ( monotonicity ) the control function is monotonically decreasing: Γ( t ) ≥ Γ( t (cid:48) ) ∀ t (cid:48) > t Condition (i) is simply that the system is initially prepared in the ground state ofthe driver Hamiltonian. This condition is necessary for AQC, and is also standardfor QW. Condition (ii) prevents pathological behaviour where the driver spectrumis effectively inverted by taking negative values of the control function Γ( t ). Thiscondition is satisfied in all traditional AQC and QW settings. Condition (iii) is thatthe quench is monotonic ; this condition excludes methods such as reverse annealing,both the dissipatively driven form proposed in Chancellor (2017) and implementedon D-Wave devices (D-Wave Systems Inc., 2019b), and the similar coherent methodproposed in Perdomo-Ortiz et al. (2011) which is sometimes also referred to as reverseannealing. The biased driver Hamiltonian proposed in Duan et al. (2013); Graß (2019)is compatible with condition (iii). Our results do not rely on the adiabatic theoremand the control function Γ( t ) does not need to be a continuous function.Without loss of generality, the driver Hamiltonian H drive can be chosen suchthat its ground-state eigenvalue is zero. Let E Γ ( t ) = (cid:104) ψ ( t ) | H Γ ( t ) | ψ ( t ) (cid:105) be theexpectation value of the energy at time t . Then, it follows immediately from condition(i) that, at time t = 0, E Γ (0) = (cid:104) ψ (0) | H prob | ψ (0) (cid:105) . (11)Furthermore, it follows from conditions (i) and (ii) that, at any later time t > E Γ ( t ) ≥ (cid:104) ψ ( t ) | H prob | ψ ( t ) (cid:105) , (12) n energetic perspective on rapid quenches in quantum annealing (cid:104) ψ ( t ) | H drive | ψ ( t ) (cid:105) ≥ (cid:104) ψ (0) | H drive | ψ (0) (cid:105) = 0 can only increase from theground state initial energy. Finally, as we show in Appendix A.2, it follows fromconditions (ii) and (iii) that the energy expectation value E Γ ( t ) monotonicallydecreases with time t ; that is, E Γ ( t (cid:48) ) ≤ E Γ ( t ) ∀ t, t (cid:48) : t (cid:48) > t (13)Taken together, the statements in (11), (12) and (13) imply (cid:104) ψ ( t = 0) | H prob | ψ ( t = 0) (cid:105) ≥ (cid:104) ψ ( t = t f ) | H prob | ψ ( t = t f ) (cid:105) , (14)for final time t f . In other words, the energy expectation with respect to theproblem Hamiltonian can only decrease compared with the initial state. If the energyexpectation of the problem Hamiltonian decreases, then the probability of measuringlow energy states increases.Appendix A.2 shows (14) holds for quenches, parameterized with the singlecontrol function Γ( t ), in the form of (2). However, since the control function Γ( t )is identified with the ratio A ( t ) / B ( t ) of control functions for quenches in the form of(1), the result in (14) follows automatically for quenches in A ( t ), B ( t ) form, exceptfor when B (0) = 0, when Γ(0) is not well-defined. In Appendix A.3, we extend to thecase where B (0) = 0, with the additional condition that the driver Hamiltonian H drive has a finite gap between its ground and first-excited manifolds (which is automaticallytrue for all Hamiltonians on Hilbert spaces of finite dimension).The key result is that, for quenches where the control function Γ( t ) decreasesmonotonically, the energy expectation value of the problem Hamiltonian H prob cannotbe higher than its initial value. Put another way, on average, a monotonic quenchcan never perform worse than random guessing. This result is important for tworeasons. Firstly, although not being harmful to average solution quality is a ratherweak statement, it applies very generally to a broad class of algorithms. Secondly,and more importantly, this result can be built upon to determine control functionsthat can provide a significant improvement, which is important for algorithm design.To do this, we need to combine the result in this section with criteria for when thetransfer of amplitude between computational basis states will be significant, which weobtain in the next section.
4. Ensuring significant dynamics
In section 2, we showed examples of a quantum quench giving significantly betterperformance than pure quantum walks. In this section, we consider theoretically howa significant improvement can occur. We know from section 3 that dynamics will neverbe detrimental; this means that, if dynamics occur, in general it will be beneficial.What remains is to determine the circumstances in which significant dynamics willoccur.
In the analytical solutions for unstructured search in a continuous-time setting (Rolandand Cerf, 2002; Childs and Goldstone, 2004), the method involves analysing thedynamics in a two dimensional subspace. To obtain significant dynamics in thissetting, the hopping rate γ or schedule functions A ( t ), B ( t ) must carefully balance therelative strengths of the driver and problem Hamiltonians, such that the off-diagonal n energetic perspective on rapid quenches in quantum annealing | j (cid:105) and | k (cid:105) connected by thedriver, i.e., (cid:104) j | H drive | k (cid:105) (cid:54) = 0, and define an effective two-level system Hamiltonian H ( jk )Γ ( t ) = Γ( t ) H ( jk )drive + H ( jk )prob (15)with the local problem Hamiltonian H ( jk )prob defined as H ( jk )prob = (cid:18) E ( j ) E ( k ) (cid:19) , (16)where E ( j ) = (cid:104) j | H prob | j (cid:105) is the energy of computational basis state | j (cid:105) with respectto the problem Hamiltonian (similarly for k ), and with the local driver Hamiltonian H ( jk )driver defined as H ( jk )drive = (cid:18) (cid:104) j | H drive | j (cid:105) (cid:104) j | H drive | k (cid:105)(cid:104) k | H drive | j (cid:105) (cid:104) k | H drive | k (cid:105) (cid:19) . (17)The extent to which the local subspace Hamiltonian H ( jk )Γ ( t ) can transfer amplitudebetween the basis states | j (cid:105) and | k (cid:105) can be characterised by comparing the off-diagonalenergy scale to the diagonal one. Define a local transfer coefficient , which takes values0 ≤ T ( jk ) ≤
1, as T ( jk ) = R (cid:104) H ( jk )drive , H ( jk )prob (cid:105) (18) ≡ t ) | (cid:104) k | H drive | j (cid:105) | t ) | (cid:104) k | H drive | j (cid:105) | + | ∆ jk | . (19)where ∆ jk = (cid:110) Γ( t ) (cid:104) j | H drive | j (cid:105) + E ( j ) (cid:111) − (cid:110) Γ( t ) (cid:104) k | H drive | k (cid:105) + E ( k ) ) (cid:111) is the difference between the diagonal elements in the diagonal basis of the problemHamiltonian.Similarly, as implied by the energy redistribution mechanism described in section3, transfer between driver eigenstates is also important. To capture this, we definea local driver coefficient D ( jk ) by transforming the local subspace Hamiltonian in H ( jk )Γ ( t ) into the diagonal basis of the local driver Hamiltonian H ( jk )drive and writing asimilar expression to (19). That is, D ( jk ) = R (cid:104) U ( jk ) † H drive U ( jk ) , U ( jk ) † H prob U ( jk ) (cid:105) , (20) n energetic perspective on rapid quenches in quantum annealing U ( jk ) is a unitary such that U ( jk ) † H ( jk )drive U ( jk ) is diagonal.It is easily shown that, for unbiased drivers such as (3), the local driver coefficient D ( jk ) and local transfer coefficient T ( jk ) are related by D ( jk ) = 1 − T ( jk ) . This makesit clear there is a trade off between the two quantities to obtain significant dynamicsunder the combined Hamiltonian. We quantify the overall level of amplitude transferwe expect by the product of the transfer and driver coefficients T ( jk ) and D ( jk ) , whichwe call the dynamic coefficient, Dyn ( jk ) = T ( jk ) D ( jk ) . (21)For unbiased drivers, since D ( jk ) = 1 − T ( jk ) , and 0 ≤ D ( jk ) , T ( jk ) ≤
1, it follows thatDyn ( jk ) satisfies 0 ≤ Dyn ( jk ) ≤ . ( jk ) captures the level of algorithmically useful localdynamics experienced by the system. In particular, if Γ (cid:29)
1, then the driverHamiltonian dominates and the problem Hamiltonian H prob will have little effect onthe dynamics of the system. Since the initial state is the ground state of the driverHamiltonian, the dynamics are driven by the much smaller problem Hamiltonian onshort timescales. This limit is captured by the dynamical coefficient, as D ( jk ) ≈
0, andhence Dyn ( jk ) ≈
0. In the opposite extreme, if Γ (cid:28)
1, then the problem Hamiltoniandominates, but since it is diagonal, the dynamics will consist almost entirely of phaserotations in the computational basis, and the amplitudes will change very little. Thislimit is captured by the transfer coefficient, as T ( jk ) ≈
0, and hence Dyn ( jk ) ≈ ( jk ) over the values of j and k which correspond to a non-zero offdiagonal element in H drive . That is, we define the average dynamic coefficientDyn = (cid:104) Dyn ( jk ) (cid:105) jk (22)where (cid:104) · (cid:105) jk represents the mean over all pairs of computational basis states j, k connected by the driver Hamilton H drive . Although (22) cannot be exactly calculatedefficiently, it should in general be possible to approximate it efficiently (up to additiveerror) by sampling. This follows from the fact that the values of Dyn ( jk ) are bounded0 ≤ Dyn ( jk ) ≤ .
25, and therefore the error δ Dyn can be reduced to the range thisvalue can take, multiplied by the statistical noise in the sample, which scales as thesquare root of the number of samples, i.e., δ Dyn ∼ . N / samples . (23)Equipped with the definition of the average dynamic coefficient Dyn, we caninvestigate when it is possible to find a value of Γ( t ) such that Dyn is large enough forsignificant short time dynamics to be generated. For simplicity, we restrict ourselvesto the unbiased driver case, when the local driver coefficient D ( jk ) and local transfercoefficient T ( jk ) are related by D ( jk ) = 1 − T ( jk ) . In this case, the local dynamiccoefficient Dyn ( jk ) can be written in terms of the driver strength Γ( t ) and a single scaled gap parameter ζ jk = | ∆ jk | |(cid:104) k | H drive | j (cid:105)| asDyn ( jk ) = ζ jk / Γ( t ) (1 + ζ jk / Γ( t ) ) . (24)If we write p ζ for the probability density function that governs the distribution of ζ jk in the particular problem and driver Hamiltonians under consideration, then it can be n energetic perspective on rapid quenches in quantum annealing µ /µ m a x Γ ( D y n ) Lower boundmax(Dyn) = 0 . . µ /µ = 1 . Figure 8.
Semi-analytical lower bound (solid, red) on Dyn as a functionof ratio of moments µ ( p ζ ) / µ ( p ζ ) of the distribution, governed by p ζ , of therescaled energy gaps ζ jk between computational basis states connected by thedriver Hamiltonian H drive . Also shown, minimum (0 .
0, dot-dashed, green) andmaximum (0 .
25, dotted, blue) possible values of Dyn. The lower-bound is non-trivial for µ ( p ζ ) / µ ( p ζ ) < . shown that the maximum value attained by the average dynamic coefficient Dyn forany choice of driver strength Γ( t ) has a lower bound which can be stated formally asmax Γ ( t )(Dyn) ≥ max 0, but is trivially zero otherwise.This shows that there is a continuous range where Dyn is bounded away from zero,and hence dynamics will definitely happen on short timescales, even for non-optimalchoices of Γ( t ). This bound is in general far from tight, but still allows us to producesome interesting examples. We next illustrate the calculation of Dyn and the lowerbound in (25) for some specific cases. As a simple example, consider the problem Hamiltonian H (2 Q )prob = − ˆ Z ˆ Z − 12 ˆ Z as defined in (5), with a transverse field driver as defined in (3). For this problemHamiltonian, there are four two level subspaces connected by the driver, | (cid:105) ↔ | (cid:105) , n energetic perspective on rapid quenches in quantum annealing | (cid:105) ↔ | (cid:105) , | (cid:105) ↔ | (cid:105) , and | (cid:105) ↔ | (cid:105) . Due to symmetry under exchange of thequbits the subspaces defined by | (cid:105) ↔ | (cid:105) and | (cid:105) ↔ | (cid:105) will behave identically,as will those defined by | (cid:105) ↔ | (cid:105) and | (cid:105) ↔ | (cid:105) . For | (cid:105) ↔ | (cid:105) and | (cid:105) ↔ | (cid:105) ,the scaled gap parameter is ζ jk = | ∆ jk | |(cid:104) k | H drive | j (cid:105)| = 3 / 2, while for | (cid:105) ↔ | (cid:105) and | (cid:105) ↔ | (cid:105) , we have ζ jk = 1 / 2. We can thus calculate Dyn exactly,Dyn = 12 (cid:18) / (2 Γ)(1 + 3 / (2 Γ)) + 1 / (2 Γ)(1 + 1 / (2 Γ)) (cid:19) = Γ (cid:18) + 1(1 + 2 Γ) (cid:19) , (26)where the time dependence in Γ( t ) has been omitted for clarity. To obtain themaximum value of Dyn we need to maximize the bound in (26) with respect to Γ. Thisis easiest done numerically, giving max Γ (Dyn) ∼ . 232 for Γ (cid:39) . p ζ is µ ( p ζ ) = 1, while the second momentis µ ( p ζ ) = 0 . 25. Based on the ratio µ ( p ζ ) µ ( p ζ ) = 0 . 25, we obtain the lower boundmax Γ Dyn (cid:38) . We consider the Sherrington-Kirkpatrick spin-glass problem Hamiltonian given in (6).We take the driver Hamiltonian H drive to be the transverse field defined in (3). Dueto the promising results found in Callison et al. (2019) for solving this problem withquantum walks, as well as for the more general quenches presented in section 2, weexpect intuitively that it should be generally possible to find values of Γ for which theaverage dynamic coefficient Dyn takes an appreciable value.The transverse field driver only connects pairs of states j, k that differ by a singlebit flip. Thus, it can be seen from (6) that, for all such pairs, the energy differencecan be written ∆ jk = − (cid:88) b (cid:54) = a s ( j ) ab J ab − s ( j ) a h a (27)where a is the index of the spin that is flipped between states | j (cid:105) and | k (cid:105) , the sum runsover b which indexes the other spins, s ( j ) ab is the eigenvalue ( ± 1) of the operator Z a Z b on the state | j (cid:105) and s ( j ) a is the eigenvalue ( ± 1) of the operator Z a on the state | j (cid:105) .The gaps ∆ jk in (27) is a sum of normally distributed variables with mean 0, and so∆ jk is itself a normally distributed variable with mean 0, and can be shown to have astandard deviation ς = (cid:112) n + 1) σ . Then, since (cid:104) k | H drive | j (cid:105) = 1 for the unbiasedtransverse field driver, the scaled gap ζ jk is distributed according to the half-normal distribution with probability density function p ζ ( ζ ) = 1 ς √ π exp (cid:18) − ζ ς (cid:19) , ζ ≥ µ ( p ζ ) µ ( p ζ ) = 1 − / π / π (29) ≈ . , (30) n energetic perspective on rapid quenches in quantum annealing ς of the distribution of the scaled gap ζ jk . For this value of the ratio, the lower bound shown in Fig. 8 ismax Γ (Dyn) (cid:38) . . (31)While this value is small compared to the maximum possible value of Dyn = 0 . ς of the distribution of the scaled gap ζ jk and thus doesnot scale with the system size. Bounding Dyn away from zero for all sizes provesthat dynamics will occur over short timescales for suitable control parameters, thusproviding evidence that the scaling found in Callison et al. (2019) may continue touseful problem sizes. As a contrasting example, we consider the problem of unstructured search on n qubits,in which a single computational basis state | m (cid:105) , out of the total N = 2 n basis states,is marked by being given a lower energy. The Hamiltonian for this problem is H search = − | m (cid:105)(cid:104) m | . (32)and again we take the driver Hamiltonian H drive to be the transverse field definedin (3). While unstructured search is a well known example with a provable quantumadvantage, the algorithms which yield this advantage all involve coherent operations ontime scales of order √ N = 2 n rather than the short-time dynamics we are discussingin this paper. As such, we would intuitively not expect the lower bound in (25) to belarge in this case.Of the n n − total off diagonal matrix element pairs in the transverse field driver,only n of these will connect a pair of computational basis states with non-zero energydifference, having energy difference ∆ jk = 1 , with the remaining n n − − n pairshaving zero-energy difference ∆ jk = 0. Therefore, the distribution of scaled gaps ζ jk can be written as p ζ ( ζ ) = nn n − δ ( ζ − 1) + (cid:16) − nn n − (cid:17) δ ( ζ ) (33)Calculating the first and second central moments of this distribution gives µ ( p ζ ) = 12 n − (34) µ ( p ζ ) = 12 n − − (cid:18) n − (cid:19) (35)and so the relevant ratio of moments is µ ( p ζ ) µ ( p ζ ) = 2 n − − n > n n − − n pairs of states j, k with | ∆ jk | = 0, Dyn ( jk ) = 0 ∀ Γ.For the remaining n pairs of states j, k with | ∆ jk | = 0, the choice of driver strengthΓ = 1 . ( jk ) = 0 . 25 for all remaining pairs of states. Thus, the n energetic perspective on rapid quenches in quantum annealing n − × . 25= 12 n +1 (37)which tends toward the lower bound of zero in the limit as n → ∞ .This tells us that, for search, most two-level subspaces do not exhibit dynamicsand probability enhancement of the marked state can only happen through finelytuned control. For an adiabatic algorithm, this is achieved by slowly adjusting theHamiltonian within a precise range so that the system can follow a very delicatepath, whereas for quantum walk this is achieved by reaching a finely tuned resonancebetween the marked state and the rest of a symmetric subspace of the Hilbert space.While interpolations between these two extremes are possible (Morley et al., 2019),all of the interpolated algorithms also rely on dynamics of a two level system with agap proportional to √ N = 2 n . In such a system, significant dynamics cannot occurin the timescales of rapid quenches, O (1) or O (poly( n )). 5. Using dynamics to find heuristic quench parameters As mentioned in section 4, the average dynamic coefficient Dyn can in general beefficiently estimated by sampling. In this section, we show via two practical examplesthat this estimate can be used to develop heuristic methods for setting the controlfunction Γ( t ), or equivalently, A ( t ) and B ( t ), for a rapid quench, in both quantumwalk and quantum annealing settings. In both cases, we use the unbiased transversefield driver Hamiltonian defined in (3). First, we consider the quantum walk algorithm,starting with a simplified example of a two qubit system. We then develop a heuristicfor the Sherrington-Kirkpatrick spin-glass, and show that it performs almost as well asthe numerically fine-tuned heuristic described in Callison et al. (2019), without needingany fine-tuning. Second, we develop a simple heuristic method for defining a schedulefor a time-dependent rapid quench, also applied to the Sherrington-Kirkpatrick spin-glass, that outperforms a linear ramp.In all the examples discussed in this section, we computed the average dynamiccoefficient Dyn numerically using all non-zero j , k pairs, rather than estimating it bysampling such pairs. This is computationally easy to do at these problem sizes, andallows us to separate the effectiveness of the heuristic from errors due to sampling. For a quantum walk, the average dynamic coefficient Dyn is a function of the chosenhopping rate Γ( t ) = γ . Informed by the result in section 3 that dynamics will typicallybe useful, it follows that by maximizing Dyn we can obtain a heuristic hopping rate γ Dyn , that should ensure significant dynamics occur over short timescales. For the twoqubit Hamiltonian from (5), Fig. 9 shows how the average success probaility within100 dimensionless time units P varies with γ . For this two qubit system, we canexactly calculate Dyn, see section 4.2, shown in Fig. 9. The maximum value of Dyngives a value for γ Dyn which is a good quality estimate for the value of γ opt . Usingbisection and a numerically calculated derivative, we find that γ Dyn ≈ . P occurs around γ = 1. Since the peak of P is quite broad, the n energetic perspective on rapid quenches in quantum annealing γ . . . . p . . . . . . D y n Figure 9. Average success probability p between t = 0 and t = 100 (blue,solid), calculated based on 10 , 000 independent random points within this rangeand Dyn (red, dashed) versus γ for the two qubit system given in (5). Dottedvertical line indicates the value of γ Dyn . n − − − − − − › P s h o r t fi γ Dyn γ heur Figure 10. Log-linear plot of average short time success probability (cid:104) P short (cid:105) against number of qubits n for quantum walks on the spin-glass dataset fromChancellor et al. (2019), using the heuristic hopping rate γ dyn derived for eachinstance by optimizing the average dynamic coefficient Dyn (red). Also shown forcomparison, (cid:104) P short (cid:105) obtained using the fine-tuned heuristic hopping rate γ heur (blue) described in Callison et al. (2019). discrepancy between γ Dyn and γ opt only reduces P by a small amount, as can beseen in Fig. 9.To test how well this heuristic hopping rate works for a more realistic example,we numerically calculated γ Dyn for each instance of size 5 ≤ n ≤ 15 of the spin glassproblems from Chancellor et al. (2019). This was done by performing a bisectionoptimization to maximise the value of Dyn as a function of γ for each instance.Following the methods in Callison et al. (2019), we performed a short-time quantumwalk and calculated the success probability P short , which is time-averaged over a short n energetic perspective on rapid quenches in quantum annealing shorttime success probability (cid:104) P short (cid:105) = . √ n (cid:90) . √ n d t P ( t ) , (38)defined in Callison et al. (2019)), for measuring the problem ground-state. Thisis shown (red line) for each size in Fig. 10. Included for comparison (blue line)are the results from Callison et al. (2019) using the fine-tuned heuristic γ heur defined there, using properties of the eigenvalue distribution for the spin glassproblem Hamiltonian. It can be seen that, despite γ heur being numerically fine-tuned specifically for the Sherrington-Kirkpatrick spin-glass problem, it performs onlymarginally better the general method we have used here. Fitting the data produces (cid:104) P short (cid:105) ∼ O ( N ( − . ± . ) for γ heur compared to (cid:104) P short (cid:105) ∼ O ( N ( − . ± . ) for γ Dyn . The eigenvalue distribution used in Callison et al. (2019) would not generallybe available to calculate γ for real problems; this comparison shows that using Dyn isa viable method for determining a useful value for γ in this case.For the small size instances we are using, we have used all the values of Dyn ( jk ) to calculate the average in the definition of Dyn in (22). We can show that the errorin γ Dyn due to sampling a subset of Dyn ( jk ) values stays manageable for larger sizes.Consider a small error δγ in γ . Doing a Taylor expansion of Dyn( γ ) around its peakvalue Dyn max gives δ Dyn = Dyn max − Dyn( γ Dyn + δγ ) = − ( δγ ) ∂ Dyn( γ ) ∂γ (cid:12)(cid:12)(cid:12)(cid:12) γ = γ Dyn + O (cid:0) ( δγ ) (cid:1) , (39)where γ Dyn is the value of γ our heuristic would find § with the exact Dyn max . Usingthe sampling error in Dyn from (23) and rearranging yields δγ ∝ N − sample (cid:32) − ∂ Dyn( γ ) ∂γ (cid:12)(cid:12)(cid:12)(cid:12) γ = γ Dyn (cid:33) − . (40)This is a general expression that can be used for any problem Hamiltonian. For theSherrington-Kirkpatrick spin glass, we can use the distribution of the scaled gaps from(28), and the definition of Dyn ( j,k ) from (24), to obtain the average value of Dyn( γ )for SK instances, (cid:104) Dyn( γ ) (cid:105)(cid:104) Dyn (cid:105) ( γ ) = 1 ς √ π (cid:90) ∞ dζ exp (cid:26) − ζ ς (cid:27) ζ / γ (1 + ζ / γ ) . (41)Making the substitution z = ζ / (2 √ ς ) , to remove the ς dependence in the exponential,and differentiating twice w.r.t. γ gives ∂ ∂γ (cid:104) Dyn (cid:105) ( γ ) = 8 ς √ π (cid:90) ∞ dz z exp (cid:8) − z (cid:9) ( γ/ς ) − √ z { ( γ/ς ) + 2 √ z } . (42) § Note that the first order term is absent, and the second order term is negative, because we aresampling at the maximum, and we have implicitly assumed that there are no other peaks in Dyn whichtake similar values. If ∂ Dyn( γ ) ∂γ (cid:12)(cid:12)(cid:12) γ = γ Dyn vanishes, then the next non-zero even derivative should beused. n energetic perspective on rapid quenches in quantum annealing t/t f s LinearHeuristic (a) t P LinearHeuristic (b) Figure 11. Left: A heuristic quench schedule of duration t f = 2 . t f = 2 . 0) is also shown. Right: The instantaneous success probability P ( t ) for measuring the problem ground-state for for each time t as the quenchprogresses along the heuristic schedule (red) and the linear schedule (blue). This needs to be evaluated at γ = γ Dyn , at the peak of (cid:104) Dyn (cid:105) ( γ ), which doing thesubstitution z = ζ / (2 √ ς ) in (41) shows occurs at a fixed value of γ / ς . Hence, the scalingwith n of the double derivative at γ = γ Dyn is determined solely by the ς − prefactorin (42). Recalling from section 4.3 that ς = (cid:112) n + 1) for these SK spin glasses, andputting it back into (40) we have δγ ∝ N − sample ( n + 1) / . (43)Callison et al. (2019) determined numerically that the peak in the success probabilityas a function of γ is very broad for SK spin glasses, and that the width of this peakdecreases as / n . Combined with (43), this means the sampling rate to calculate Dynneeds to increase by a poly( n ) factor as n increases, in order to determine γ Dyn tosufficient accuracy. Since n corresponds to the number of qubits, this can be doneefficiently. For a time-dependent rapid quench of the form H AB ( t ) defined in (1) and totalduration t f , a common choice of control functions, inspired by the adiabatic algorithm,is A ( t ) = 1 − s ( t ) and B ( t ) = s ( t ), where s ( t ) is a schedule function with boundaryconditions s (0) = 0 and s ( t f ) = 1. In the absence of any knowledge about wherealong the schedule useful computation can happen, the schedule function is often setto be the linear function s ( t ) = t / t f . The average dynamic coefficient Dyn provides ameasure of the level of dynamics at each point along the schedule. Intuition gainedfrom section 3 suggests that the linear schedule can in general be improved by spendingless time in regions where Dyn is small and more time in regions where Dyn is large. Astraightforward way to do this is to choose d s d t ∝ (the constant of proportionality isset by the boundary conditions s (0) = 0 and s ( T ) = 1). We have approximated such aschedule for a typical nine qubit Sherrington-Kirkpatrick spin-glass instance, as shownin Fig. 11(a) (red line). We have done this by fixing the value of the points marked n energetic perspective on rapid quenches in quantum annealing s ∝ ∆ t Dyn , subject to the boundary conditions, and thenlinearly interpolating between them. A linear schedule s ( t ) = t / t f (blue line) is alsoshown for comparison. Figure 11(b) shows the instantaneous success probability P ( t )for measuring the problem ground-state as the quench progresses along the heuristicschedule (red line) and the linear schedule (blue line) for quench duration of t f = 2.It can be seen that the simple heuristic we’ve used here has resulted in a significantimprovement in success probability at the end of the schedule. We have checkedsufficiently many of the instances to determine that this level of improvement is typicalfor this size of problem and total time duration t f = 2. Further improvements maybe available by varying t f or choosing a different function of Dyn for d s d t . 6. Numerical methods Numerical simulation and optimization were used extensively throughout this work, asmuch of the analysis we have performed is not analytically tractable. The simulationsand plots were performed using the Python language (Van Rossum and Drake, 2003),aided extensively by the NumPy (Oliphant, 2006), SciPy (Jones et al., 2001–), quimb,(Gray, 2018), and Matplotlib (Hunter, 2007) libraries. We also used the IPythoninterpreter (P´erez and Granger, 2007) and Jupyter notebook system (Kluyver et al.,2016). MATLAB was used for some early numerical experiments, but not for anyresults which directly appear in the manuscript.The numerical optimization used to produce Figs. 8, 9, 10 and 11, as well as thecurve fitting used in Figs. 7 and 10, was performed using the optimization tools inSciPy (Jones et al., 2001–).The Sherrington-Kirkpatrick spin glass instances in the data repository atChancellor et al. (2019) have been used extensively. In any cases where asingle example Sherrington-Kirkpatrick spin-glass instance has been used, it is theinstance ovcjhwbhtcpcvwicoxpdpvjzqojril . The plot of average short time successprobability (cid:104) P short (cid:105) against number of spins n in Fig. 10 uses all of the Sherrington-Kirkpatrick spin glass instances in the repository. 7. Summary and further work In this paper, we have generalised and extended work begun in Callison et al. (2019) totime-varying quantum annealing schedules. Callison et al. (2019) provide numericalevidence for the ability of quantum walks to solve NP hard problems using manyrepeats of short runs. This strategy scales better than quantum search, by exploitingthe correlations in the problem Hamiltonian. The energy conservation mechanismidentified in Callison et al. (2019) explains how energy conserving quantum walkscan find lower energy states with better than guessing probability. In section 3, wegeneralised the energy conservation mechanism to an energy redistribution mechanism that holds for all monotonic quenches which start in the ground state of the driverHamiltonian and have non-negative control functions.The improvements leveraged by time-varying rapid quenches can be considerable,as we illustrated in section 2. To generate significant energy redistribution, thereneeds to be significant dynamics driving the system away from the initial state. Tocharacterise the dynamics, in section 4 we defined the average local dynamic coefficientthat balances the contributions from both the driver and problem Hamiltonians. This4allows the control functions in the Hamiltonian to be optimised for fast dynamics, andprovides a very general way to estimate good values to use for specific problems. Forthe spin glass data (Chancellor, 2019), we showed in Fig. 10 that such estimates arealmost as good as the numerically optimised values used in Callison et al. (2019).Taken together, the energy redistribution mechanism and the average dynamiccoefficient are powerful tools for understanding, designing, and optimally controllingrapid quench quantum annealing algorithms. While adiabatic quantum computingand quantum walk search have long had theoretical underpinnings, this represents asignificant step in understanding how to exploit quantum annealing schedules run forshort times. For current state-of-the-art noisy quantum computers, short run timesare a big advantage over the long coherence times required for adiabatic quantumcomputing, or quantum walk search.We have shown that our tools apply to the biased drivers proposed in Duanet al. (2013); Graß (2019), which provide a method of incorporating prior informationinto annealing schedules. This can produce significant improvements, as we illustratein section 2.2. On the other hand, reverse annealing schedules, both as proposedby Perdomo-Ortiz et al. (2011); Ohkuwa et al. (2018); Yamashiro et al. (2019), andas implemented in the latest D-Wave Systems (D-Wave Systems Inc., 2019b), areby definition not monotonic, so the tools and mechanisms identified here cannot beapplied. Since reverse annealing is a powerful tool, extending our results to non-monotonic cases is a worthwhile direction for further research. Acknowledgements We thank Jemma Bennett for providing useful references related to error correction.NC and JC were supported by EPSRC grant EP/S00114X/1. LN was supported by aDurham University studentship. AC was funded by EPSRC grant EP/L016524/1 viathe Imperial College London CDT in Controlled Quantum Dynamics. VK and NCwere supported by EPSRC grant EP/L022303/1. Appendices A. Proof: monotonic quenches do no worse than guessing A.1. Energy conservation mechanism We first recap the special case presented in Callison et al. (2019); Hastings (2019)for time independent controls. Quantum walks can be viewed as a closed-systemannealing protocol with a discontinuous schedule (Morley et al., 2019). For QW,when formulated in terms of Eq. (1) A ( t ) and B ( t ) are constant, independent of time.This picture however doesn’t follow the convention of how annealing protocols areformulated, where the system starts in the ground state of the initial Hamiltonian andthe driver is completely absent at the end of the anneal. Following such a convention isimportant for instance to define an interpolation between annealing protocols and QW,as was done in (Morley et al., 2019). To define QW as an annealing protocol in which A (0) = B ( t fin ) = 1 and A ( t fin ) = B (0) = 0, we can write A ( t ) = γ Θ( t fin − t + (cid:15) ) and B ( t ) = Θ( t − (cid:15) ), where Θ is the Heaviside theta function, Θ( a > 0) = 1, Θ( a < 0) = 0,Θ( a = 0) = ,and take the limit where (cid:15) → | ψ ( t = 0) (cid:105) is a ground state of the driver Hamiltonian H drive , it follows immediately that the expectation value of the driver Hamiltonianis at its lowest at t = 0, that is, (cid:104) ψ ( t ) | H drive | ψ ( t ) (cid:105) ≥ (cid:104) ψ ( t = 0) | H drive | ψ ( t = 0) (cid:105) ,since the expectation value of the driver Hamiltonian H drive for any quantum statecannot be less than that of the ground state.The total energy expectation as a function of time can be written E ( t ) = (cid:104) ψ ( t ) | γH drive + H prob | ψ ( t ) (cid:105) = γ (cid:104) H drive (cid:105) ψ ( t ) + (cid:104) H prob (cid:105) ψ ( t ) , (44)where the notation (cid:104) . (cid:105) ψ is used to denote the expectation value with respect to thestate ψ has been adopted. Since energy is conserved for 0 < t < t fin , it follows that,for (cid:15) → E ( (cid:15) ) = E ( t f − (cid:15) ), and therefore γ (cid:104) H drive (cid:105) ψ ( t =0) + (cid:104) H prob (cid:105) ψ ( t =0) = γ (cid:104) H drive (cid:105) ψ ( t f ) + (cid:104) H prob (cid:105) ψ ( t f ) (45)rearranging terms, and recalling that ψ ( t = 0) is the ground state of H drive and γ ≥ (cid:104) H prob (cid:105) ψ ( t f ) − (cid:104) H prob (cid:105) ψ ( t =0) = γ [ (cid:104) H drive (cid:105) ψ ( t =0) − (cid:104) H drive (cid:105) ψ ( t f ) ] ≤ , (46)and therefore (cid:104) H prob (cid:105) ψ ( t f ) ≤ (cid:104) H prob (cid:105) ψ ( t =0) . Since ψ ( t = 0) is not an eigenstateof the full Hamiltonian, some dynamics are guaranteed to happen, and thus therewill be times t > (cid:104) H prob (cid:105) ψ ( t ) is strictly less than (cid:104) H prob (cid:105) ψ ( t =0) . In thenext subsection we show that by discretizing and using a rescaled representation,we can use a combination of the energy conservation mechanism and arguments aboutthe stages where the Hamiltonian is modified, to generalize a version of the energyredistribution mechanism to all cases of time dependent Hamiltonian evolution whichobey the conditions given in section 3. A.2. Energy redistribution mechanism We aim to show that the energy expectation value E Γ ( t ) = (cid:104) ψ ( t ) | H Γ ( t ) | ψ ( t ) (cid:105) (47)decreases monotonically with time. To do so, we first introduce a discretizedapproximation to the evolution as | ψ ( q ) k (cid:105) = T (cid:89) k (cid:48) = k exp( − iH Γ ( k (cid:48) t f q ) t f q ) | ψ (0) (cid:105) , (48)for 1 ≤ k ≤ q and where the symbol T is added to emphasise that the time order ofthe product must be preserved, since the Hamiltonians at different times are non-commuting. This discretized approximation becomes exact in the limit q → ∞ .The evolution of a quantum system under the time dependent Hamiltonian givenin (1) from time t = 0 to time t = t f from the initial state | ψ (0) (cid:105) is broken downas follows: The initial state is evolved under the constant Hamiltonian H ( t f q ) fortime t f q to produce a state | ψ ( q )1 (cid:105) which then evolves under the constant Hamiltonian H (2 t f q ) for time t f q and so on, until a final state | ψ ( q ) q (cid:105) is reached. Then, in the limit, | ψ ( t f ) (cid:105) = lim q →∞ | ψ ( q ) q (cid:105) . This kind of discretization can be thought of as an extension6of the Suzuki-Trotter decomposition (Trotter, 1959; Suzuki, 1993) and is thereforesometimes informally referred to as Trotterization. In the same manner, we can definea discretized version of the energy expectation value as E ( q )Γ ,k = Γ (cid:18) k (cid:48) t f q (cid:19) (cid:68) ψ ( q ) k | H drive | ψ ( q ) k (cid:69) + (cid:68) ψ ( q ) k | H prob | ψ ( q ) k (cid:69) , (49)Quantum states are only defined up to a constant phase, which is equivalent tochoosing an arbitrary ‘zero’ for the energy. Hence, without loss of generality, we chooseto set (cid:104) ψ (0) | H drive | ψ (0) (cid:105) = 0; in other words, we impose semidefiniteness on H drive by defining its ground state | ψ (0) (cid:105) to have eigenvalue 0.During each time-independent evolution step, the energy expectation value E ( q )Γ ,k is conserved. Furthermore, since by definition H drive is positive semidefinite andΓ( ( k +1) t f q ) ≤ Γ( kt f q ), it follows that E ( q )Γ ,k +1 ≤ E ( q )Γ ,k . (50)Repeated application of this inequality results in the more useful inequality E ( q )Γ ,q ≤ E ( q )Γ , . (51)Since E ( q )Γ , is the rescaled energy during the whole of the first evolution step, it followsthat E ( q )Γ , = E Γ ( t = 0) . (52)Furthermore, we have that lim q →∞ E ( q )Γ ,q = E Γ ( t f ) . (53)which means E Γ ( t f ) ≤ E Γ (0) . (54)Since this equation holds for all t f > 0, we have shown that E Γ ( t ) monotonicallydecreases with t . A.3. The case of B ( t ) → : divergence of ΓThe result in section 3 is that the inequality (14) holds for any quench with aHamiltonian in the form of (2) that satisfies the three conditions listed in section 3. Wenow consider quenches with a Hamiltonian in the form of (1). Any Hamiltonian of theform (1) with B (0) > A ( t ) / B ( t ) with Γ( t ) and rescaling by a factor / B ( t ) , which can be formally compensated for byrescaling time by a factor of B ( t ). Thus, the inequality (14) holds also for any quenchwith a Hamiltonian in the form of (1) with B (0) > B (0) = 0.In the case that B (0) = 0, consider the modified Hamiltonians H (cid:48) drive = H drive − (cid:15)A (0) H prob (55) H (cid:48) prob = H prob (56)7and the modified control functions A (cid:48) ( t ) = A ( t ) (57) B (cid:48) ( t ) = B ( t ) + A ( t ) A (0) (cid:15) = B ( t ) (cid:20) t ) (cid:15)A (0) (cid:21) , (58)where (cid:15) (cid:28) 1. It can be seen that that total Hamiltonian is unchanged, H (cid:48) A,B ( t ) ≡ A (cid:48) ( t ) H (cid:48) drive + B (cid:48) ( t ) H (cid:48) prob = A ( t ) H drive + B ( t ) H prob , (59)but we have that B (cid:48) (0) = (cid:15). (60)We define Γ (cid:48) ( t ) ≡ A (cid:48) ( t ) B (cid:48) ( t ) (61)Γ (cid:48) ( t ) = Γ( t ) (cid:104) t ) (cid:15)A (0) (cid:105) . (62)It can be immediately seen that Γ (cid:48) ( t ) is non-negative if Γ( t ) is non-negative, and socondition (ii) is satisfied. Furthermore,dΓ (cid:48) ( t )dΓ( t ) = 1 (cid:104) t ) (cid:15)A (0) (cid:105) . (63)Thus, Γ (cid:48) ( t ) is monotonically-decreasing if Γ( t ) is is monotonically decreased, and socondition (iii) is satisfied.If we were to start the protocol in the state | ψ (cid:48) gs (cid:105) , a ground-state of H (cid:48) drive ,condition (i) would be satisfied and the result would be proven. However, the originalprotocol we are considering starts in the the state | ψ (0) (cid:105) , ground-state of H drive .Applying first order perturbation theory in (cid:15) to H (cid:48) drive , we find that H (cid:48) drive has aground-state | ψ (cid:48) gs (cid:105) = | ψ (0) (cid:105) + O (cid:18) (cid:15)A (0)∆ (cid:19) | ψ ⊥ (cid:105) (64)where | ψ ⊥ (cid:105) is a normalized state vector orthogonal to | ψ (0) (cid:105) and ∆ is the energy gapbetween the ground and first-excited manifolds of the actual driver Hamiltonian H drive .Thus, assuming the driver Hamiltonian H drive is not gapless (which is automaticallytrue for all Hamiltonians on Hilbert spaces of finite dimension), the inequality in (14)is satisfied in the limit as (cid:15) → B. Lower bound on the average dynamic coefficient B.1. Bound on probabilities in a range based on second moment Here, we prove a useful bound that will be applied in the following subsection. Assumethat the distribution p ( x ) has a finite second moment µ ( p ) = (cid:90) ∞−∞ d x p ( x )( x − µ ( p )) , (65)8where µ ( p ) = (cid:90) ∞−∞ d x p ( x ) x, (66)is the first moment (mean). Let us choose some values x max > x min such that µ ( p ) = ( x max + x min ). The distribution q ( x ) = δ ( x min − (cid:15) ) + δ ( x max + (cid:15) ) has theminimum possible second moment while having no support in the interval [ x min , x max ],where δ is the Dirac delta distribution. In the limit (cid:15) → 0, the second moment of thisdistribution is µ ( q ) = ( x max − x min ) . Thus, if µ ( p ) < µ ( q ), then p ( x ) must havesome support within the range [ x min , x max ]. In particular, because second moment µ ( p ) can be lower bounded as µ = (cid:90) ∞−∞ d x p ( x )( x − µ ( p )) ≥ (cid:90) x min −∞ d x p ( x )( x − µ ( p )) + (cid:90) ∞ x max d x p ( x )( x − µ ( p )) ≥ µ ( q ) (cid:18)(cid:90) x min −∞ d x p ( x ) + (cid:90) ∞ x max d x p ( x ) (cid:19) = µ ( q ) (cid:18) − (cid:90) x max x min d x p ( x ) (cid:19) , the probability for x to be in the interval [ x min , x max ] can also be lower bounded as (cid:90) x max x min d x p ( x ) ≥ − µ ( p ) µ ( q )= 1 − µ ( p )( x max − x min ) (67) B.2. A simple lower bound Let ζ jk = (cid:12)(cid:12)(cid:12) ∆ jk (cid:104) H drive (cid:105) jk (cid:12)(cid:12)(cid:12) and let η jk = ζ jk Γ . Furthermore, let p ζ and p η be probabilitydensity functions that govern the distribution of the values ζ jk and η jk , respectively,over a set of problem instances. Let µ ( p ) and µ ( p ) refer to the first and secondmoments, respectively, of a distribution governed by the probability density function p . The dynamic coefficient isDyn ( jk ) = η jk (1 + η jk ) , (68)so we will consider the function f ( x ) = x (1 + x ) (69)where x > x be distributed according to the probability density function p η . We know9that the expectation value (cid:104) f ( x ) (cid:105) x is then (cid:104) f ( x ) (cid:105) x = ∞ (cid:90) d xp η ( x ) f ( x )= x max (cid:90) x min d xp η ( x ) f ( x ) + x min (cid:90) d xp η ( x ) f ( x ) + ∞ (cid:90) x max d xp η ( x ) f ( x )= P η ( x min < x < x max ) (cid:104) f ( x ) (cid:105) x max x min + P η ( x ≥ x min ) (cid:104) f ( x ) (cid:105) x min + P η ( x max ≥ x ) (cid:104) f ( x ) (cid:105) ∞ x max (70)where x max > x min , P η ( . . . ) is the probability of its argument being true if η isdistributed according to p η , and (cid:104) f ( x ) (cid:105) ba is the expectation value of f ( x ) if x isdistributed according to a (renormalized) version of p η with all support on x < a and x > b removed. 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