On Calculating the Dynamics of Very Large Quantum Systems
OOn Calculating the Dynamics of Very Large Quantum Systems
J.J. Bowen, V.M. Dwyer, I.W. Phillips, and M.J. Everitt ∗ Quantum Systems Engineering Research Group, Loughborough University, Leicestershire LE11 3TU, United Kingdom (Dated: Tuesday 7 th February, 2017)Due to the exponential growth of the state space of coupled quantum systems it is not possible,in general, to numerically store the state of a very large number of quantum systems within aclassical computer. We demonstrate a method for modelling the dynamical behaviour of measurablequantities for very large numbers of interacting quantum systems. Our approach makes use of asymbolic non-commutative algebra engine that we have recently developed in conjunction withthe well-known Ehrenfest theorem. Here we show the possibility of determining the dynamics ofexperimentally observable quantities, without approximation, for very large numbers of interactingharmonic oscillators. Our analysis removes a large number of significant constraints present inprevious analysis of this example system (such as having no entanglement in the initial state). Thismethod will be of value in simulating the operation of large quantum machines, emergent behaviourin quantum systems, open quantum systems and quantum chemistry to name but a few.
Obtaining information for large numbers of interactingquantum systems is in general a hard problem. This isdue to the fact that the Hilbert/state space describing thetotal system grows exponentially in size with the numberof its constituents. There are many open questions thatarise from this behaviour in areas such as open quantumsystems, emergent phenomena such as quantum phasetransitions and the quantum to classical crossover. Be-yond basic science, the emergence of new quantum tech-nologies demands computer aided tools to serve a similarpurpose to those, for example, in the semiconductor in-dustry (see [1–4]) and may be expected to range fromsimple quantum circuit modellers (a quantum analogueof Spice) to more sophisticated quantum VLSI applica-tions. The programming languages most often deployedwithin the physical sciences are not well equipped to dealwith the non-commuting algebra of quantum systemsthat such engineering tools would need to accommodate.We note that, while some basic functionality has recentlybeen introduced into Maple, it is not yet well suited tothe solution of complex problems. For this reason weare developing a non-commuting symbolic algebra pack-age using a functional programming language (Haskell)that we have named QuantAl (for Quantum Algebra).In this work we report our first application of QuantAlin the study of the exact dynamics of a very large set ofinteracting quantum systems. We will discuss, in detail,the form and operation of this package in a dedicatedfollow-up paper. Here we note that it is based on usingan N -ary tree data structure to represent mathematicalexpressions, using recursion to traverse through the tree,applying pattern matching to perform operations, andrespecting non-commutativity of operators within the ex-pressions. In the current work we apply this techniqueto enable the modelling and simulation of large interact-ing quantum systems, observing that in some cases thiscan be done without approximation. In addition to bet-ter understanding the physics of many body quantumsystems there are a number of technological applications which readily arise from this process including: designfor test; verification, certification and validation; systemdesign and parameter estimation.In studying large numbers of interacting quantum sys-tems, in the absence of analytical solutions, some com-promise is always necessary. Often, for example, modelsare produced that provide approximate information foronly part of the system (the reduced density operatorfrom a master equation). In this work we take a differ-ent, complementary, approach and seek limited informa-tion on the complete dynamics, through monitoring theexpected values of observable quantities. This greatlysimplifies the analysis as it then becomes unnecessaryto represent the quantum state at all. One possible ap-proach begins by considering one of the oldest results ofquantum theory - Ehrenfest’s Theorem [5], and then usesto advantage symbolic computer algebra to automate theprocess of generating the large, but often tractable, num-ber of the equations of motion needed to determine thedynamics of the expectation values.The expectation value, for any quantum mechanicaloperator ˆ A , is related to its commutation relation with asystem’s Hamiltonian ˆ H according to [5]:dd t (cid:68) ˆ A (cid:69) = − i (cid:126) (cid:68)(cid:104) ˆ A, ˆ H (cid:105)(cid:69) + (cid:42) ∂ ˆ A∂t (cid:43) (1)For one dimensional systems of position ˆ q and momentumˆ p , whose Hamiltonian with potential energy V (ˆ q ), isˆ H = ˆ p m + V (ˆ q ) (2)Ehrenfest’s theorem, when applied to ˆ q and ˆ p in turn,yields the following well-known resultdd t (cid:104) ˆ q (cid:105) = (cid:104) ˆ p (cid:105) m and dd t (cid:104) ˆ p (cid:105) = − (cid:28) ∂V (ˆ q ) ∂ ˆ q (cid:29) . (3)These two coupled equations are of the same form asthe classical equation of motion for position and momen-tum of the same system. We also observe the obvious a r X i v : . [ qu a n t - ph ] F e b fact that these two equations form a closed set of firstorder differential equations. In 1998 Nicholas Wheelerdiscussed at length, in an unpublished essay, some usefulconsequences of the Ehrenfest’s theorem including morecomplicated one-dimensional systems and a useful dis-cussion on the momental hierarchy supported by arbi-trary observables [6]. Within our method any truncationof the momental hierarchy may introduce an approxi-mation. If the hierarchy is finite, the method is exact;and in many cases will not suffer exponential scaling ofthe Hilbert space of composite systems. If the momentsdo not converge the availability of this semi-analyticalmethod makes other options practically available such asthe following. If we note that, using the evolution oper-ator, (cid:68) ψ ( t ) (cid:12)(cid:12)(cid:12) ˆ A (cid:12)(cid:12)(cid:12) ψ ( t ) (cid:69) = (cid:68) ψ (0) (cid:12)(cid:12)(cid:12) e i ˆ Ht (cid:126) ˆ Ae − i ˆ Ht (cid:126) (cid:12)(cid:12)(cid:12) ψ (0) (cid:69) . together with the Baker-Campbell-Hausdorff formula ex-pansionit is simple to form a power series solution (cid:68) ˆ A (cid:69) ( t ) = (cid:68) ˆ A (cid:69) t =0 + i t (cid:126) (cid:68)(cid:104) ˆ H, ˆ A (cid:105)(cid:69) t =0 + (cid:0) i t (cid:126) (cid:1) (cid:68) [ ˆ H, [ ˆ H, ˆ A ]] (cid:69) t =0 + . . . (4)which is accurate if computed to sufficiently high orders.This alternative method will be explored in a later pub-lication. In this work we will make use of Ehrenfest’stheorem for a system for which there are no momentalhierarchy issues and for which the method is very effi-cient - namely that of a set of coupled simple harmonicoscillators.Before proceeding to this example we outline the pro-cess of applying Ehrenfest’s theorem. To determine theequation of motion for (cid:68) ˆ A (cid:69) we are required to not justapply Eq. (1) once, but recursively to all those expecta-tion values that arise from (cid:68)(cid:104) ˆ A, ˆ H (cid:105)(cid:69) and (cid:68) ∂ ˆ A/∂t (cid:69) . Wemust continue this process, stopping only when a closedset of equations is formed or some sufficient approxima-tion is achieved (or a sufficient momental hierarchy is es-tablished). Once this is done we will have obtained a setof first order ordinary differential equations that can besolved to compute system dynamics using standard toolssuch as Matlab, and thus require only the systems ini-tial conditions. The setting of these is trivial when thestate is separable; where only small subsets of its con-stituents are entangled, or when analytical methods canbe deployed. For example, expectation values of maxi-mumly entangled coherent states are easy to determineso long as the system’s observables may be representedin terms of normally ordered creation and annihilationoperators. Given that time-dependent Hamiltonians donot necessarily greatly complicate this method, it shouldalso be possible to include more complex states as initialconditions when they are known to arise as the outcome of another dynamical process, such as through gate op-erations.Our example system comprises a set of harmonic oscil-lators coupled, in the usual way, via the position degreesof freedom. The total Hamiltonian in this case isˆ H = N (cid:88) i =1 (cid:126) ω i (cid:18) ˆ n i + 12 (cid:19) + (cid:88) i FIG. 1. Here we show the dynamics of the expectation valueof position (cid:104) ˆ q (cid:105) for a simple harmonic oscillator staring in acoherent state | α = 1 (cid:105) coupled to a bath of oscillators eachstaring in their own ground state | α = 0 (cid:105) . The top figure(a) shows the oscillator’s dynamics when the bath comprises100 oscillators and the bottom plot (b) when it contains 200.In line with expectations, it is evident that a bath of 100oscillators results in an earlier revival than for one comprising200 oscillators. The inset in (a) shows the form of the spectraldistribution of the bath that we have used to produce bothplots. that this approach has over the Master Equation methodis that the system plus environment may be started inan entangled state. This capability opens up new di-rections for study in quantum thermodynamics, out-of-equilibrium open quantum systems and high temperaturequantum phenomena – making more systems amenableto study, such as indicated in [9].In this work we have combined the well-knownEhrenfest theorem with our automated symbolic non-commutative algebra package to generate the equations D i m e n s i o n l e ss p o s i t i o n (cid:104) ˆ q (cid:105) Dimensionless time ω t (a) D o m i n a n t (cid:104) ˆ a (cid:105) o s c ill a t i o n f r e q u e n c y / ω Bath mode (b) FIG. 2. In (a) we show the dynamics of the expectationvalue of position (cid:104) ˆ q (cid:105) as for Fig. 1 (a) but now with the systemstarting off in the entangled state (cid:12)(cid:12) ψ ( ξ = 1 , ζ = , δ = ) (cid:11) asdefine in Eq. (6). The dynamics of the j th bath mode (cid:104) ˆ q j (cid:105) isdominated by oscillation at a frequency shown by the markersin (b). The parameter choices determine the number of modesoscillating at the system frequency ω , with the remainderoscillating at their own bath frequency. Linear sampling ofthe spectral density J ( ω ) has been used here after [8]. of motion for the expectation values of observables insmall to very large ensembles of quantum systems. Ourmethodology demonstrates a new paradigm for the mod-elling of such quantum systems, enabling exploration,within a fully quantum mechanical framework, of thephysics of many systems in a computationally tractable way. The example considered has an impact alreadyfor the study of Quantum Brownian Motion. It is well-known that the universal assumption of a separable ini-tial state is inadequate and leads to “slips” in the reduceddensity matrix dynamics [7]. Here we show that, for asystem oscillator tuned to a bath with high quality factor,different entangled initial states lead to different dynam-ics for both the system and the bath. Dependent on theinitial state, distinct sections of the bath oscillate freely,whilst others are synchronised to the system frequency.Our approach greatly widens not just the range of possi-ble initial states and coupling strengths, but also possibletypes of coupling and topology as well as for the inclu-sion of time-dependent Hamiltonians. As a consequence,it will enable the study of large ensembles of physicalsystems in a way that was not previously possible.The Ehrenfest theorem can be used to determine ob-servables such as (cid:104) ˆ q (cid:105) ( t ), but also a number of importantquantities such as intensity or correlation coefficients in-cluding g (which may reveal quantum effects such asanti-bunching). It may even be possible to reconstructthe dynamics of the quantum state itself in phase space asthe Wigner function (and from this the density matrix)can be written as the expectation value of a displacedparity operator (although momental hierarchy may be-come an issue here). Future applications of this methodwill lead to new insights in the areas of: open quantumsystems, emergent behaviour and the quantum to classi-cal crossover, and quantum engineering (including designfor test; verification, certification and validation; systemdesign and parameter estimation). ∗ [email protected][1] J. Gough and M. R. James, Communications in Mathe-matical Physics , 1109 (2009).[2] J. Combes, J. Kerckhoff, and M. Sarovar, arXiv preprintarXiv:1611.00375 (2016).[3] J. E. Gough, M. R. James, and H. I. Nurdin, Phys. Rev.A , 023804 (2010).[4] J. Gough and M. R. James, IEEE Transactions on Auto-matic Control , 2530 (2009).[5] P. Ehrenfest, Zeitschrift f¨ur Physik , 455 (1927).[6] N. Wheeler, “Remarks concerning the status & some ram-ifications of Ehrenfest’s theorem,” (1998).[7] C. Fleming, A. Roura, and B. Hu, Annals of Physics ,1207 (2011).[8] K. H. Hughes, C. D. Christ, and I. 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