Quantum Simulation of Molecular Collisions with Superconducting Qubits
Emily J. Pritchett, Colin Benjamin, Andrei Galiautdinov, Michael R. Geller, Andrew T. Sornborger, Phillip C. Stancil, John M. Martinis
QQuantum Simulation of Molecular Collisions with Superconducting Qubits
Emily J. Pritchett, Colin Benjamin,
1, 2
Andrei Galiautdinov, Michael R.Geller, Andrew T. Sornborger, Phillip C. Stancil,
1, 2 and John M. Martinis Department of Physics and Astronomy, University of Georgia, Athens, GA 30602 Center for Simulational Physics, University of Georgia, Athens, GA 30602 Department of Mathematics and Faculty of Engineering, University of Georgia, Athens, GA 30602 Department of Physics, University of California at Santa Barbara, Santa Barbara, California 93106 (Dated: July 8, 2010)We introduce a protocol for the fast simulation of n -dimensional quantum systems on n -qubitquantum computers with tunable couplings. A mapping is given between the control parametersof the quantum computer and the matrix elements of H s ( t ), an arbitrary, real, time-dependent n × n dimensional Hamiltonian that is simulated in the n -dimensional ‘single excitation’ subspace ofthe quantum computer. A time-dependent energy/time rescaling minimizes the simulation time onhardware having a fixed coherence time. We demonstrate how three tunably coupled phase qubitssimulate a three-channel molecular collision using this protocol, then study the simulation’s fidelityas a function of total simulation time. A quantum computer can significantly reduce the re-sources necessary to simulate quantum mechanical sys-tems [1]. Typically, quantum simulation algorithms al-gorithms construct the simulated system’s time evolu-tion operator, energies and/or eigenstates from a uni-versal set of gates [2–8]. Alternatively, ultracold atoms,trapped ions, and liquid-state NMR have directly emu-lated the time evolution of certain other quantum sys-tems [9–11]. Recent experimental progress suggests thatquantum simulation will be one of the first practical ap-plications of quantum computation [7–12].In principle, an n -qubit quantum computer can storethe state of any N = 2 n dimensional quantum system, anexponential reduction in the resources necessary to storequantum information on a classical computer. However,simulation may require ∼ N = 2 n elementary gates pertime step unless the simulated Hamiltonian has specialproperties, e.g. locality [2, 13]. Even for these specialHamiltonians, fully digital quantum simulation often re-quires an excessive number of gates for current quantumcomputing technology [4, 6].In this Letter, we show that a subspace of a tunable n -qubit quantum computer can emulate an arbitrary n -dimensional quantum system, trading an exponential re-duction in resources for simulations of a wider varietyof Hamiltonians. This subspace simulates other quan-tum systems very different from the computer itself in anamount of time that is independent of n . By comparison,classical simulation of an n -dimensional quantum systemrequires ∼ n elementary operations per time step. Whilethe most efficient quantum simulation algorithms offeran exponential reduction in both qubits and elementaryoperations, they typically apply to specific, fundamentaltime-independent Hamiltonians, or those already simi-lar to that of the computer itself. We show that with amore modest polynomial reduction in resources, a sub-space of a tunable quantum computer can simulate anyreal, time-dependent Hamiltonian. We begin by outlining the theory behind our approachto simulation. First, we identify an n -dimensional invari-ant subspace suitable for quantum simulation. Then wedefine a time dependent energy/time rescaling that max-imizes the speed of the simulation within the constraintsof the quantum computer. Finally, the control parame-ters of the quantum computer are given explicitly as afunction of the matrix elements of H s ( t ).Our approach is tested by performing a simulation ofa molecular collision with a circuit of tunably coupledJosephson phase qubits. Molecular collisions and elec-tronic structure calculations are widely studied as im-portant applications of quantum simulation techniques[6–8]. We show in detail how a superconducting circuit ofthree tunably coupled Josephson phase qubits simulatesa three channel Na-He collision. Finally, we discuss therelationship between simulation fidelity and total simu-lation time for this particular example. An n-Dimensional Subspace of the full quantum com-puter’s Hilbert space, H , can emulate another quantumsystem at all times only if it is invariant to the time evo-lution generated by the computer’s Hamiltonian H qc (sothat the subspace is well-isolated from the rest of H andevolves unitarily). We model H qc as H qc ( t ) = n (cid:88) i =1 − (cid:15) i ( t )2 σ zi + 12 (cid:88) i (cid:54) = j g ij ( t ) J µν σ µi ⊗ σ νj , (1)where (cid:15) i ( t ) are the uncoupled qubit energies, g ij ( t ) = g ji ( t ) are the pairwise qubit interaction strengths, J µν gives the relative size of the σ µi ⊗ σ νj interaction, and µ, ν ∈ { , x, y, z } are summed over. While (cid:15) i ( t ) and g ij ( t )may in general be time-dependent, the time-independentstructure of qubit interaction is specified by J µν , a di-mensionless tensor that is typically fixed by a given ar-chitecture and is identical between each pair of qubits.In the weak coupling limit, | g ij ||| J µν || /(cid:15) i (cid:28)
1, subspacesof H are invariant to time evolution generated by H qc if a r X i v : . [ qu a n t - ph ] A ug spanned by computational basis states having the samenumber of excited (tunable) qubits. The ‘single excita-tion subspace’, denoted as H n , is an n -dimensional in-variant subspace spanned by | i (cid:11) n ≡ | .. i .. n (cid:11) for all i = 1 , ..., n .The control parameters (cid:15) i ( t ) and g ij ( t ) directly controlthe Hamiltonian that H n simulates. We define H n as H qc projected into the single excitation subspace, H n ( t ) ≡ P H qc ( t ) P † (2)where P is an n × n dimensional operator that projects H onto H n . Up to an additive energy shift, H n has matrixelements H ijn ( t ) ≡ (cid:40) (cid:15) i ( t ) − α (cid:80) k (cid:54) = i g ik ( t ) , i = jg ij ( t ) , i (cid:54) = j (3)with α ≡ J zo + J zz ). We assume J xx + J yy (cid:54) = 0 andnormalize J µν so that J xx + J yy = 1. In the weak couplinglimit, H n is approximately invariant and generated by H n : U n ( t ) ≡ P U qc ( t ) P † (cid:39) T e − i (cid:126) (cid:82) t H n ( t (cid:48) ) dt (cid:48) (4)where T is the time-ordering operator. H n generates U n exactly when no matrix elements of H qc mix H n with therest of H (i.e. J x = J y = J zx = J zy = 0). The ( n + n ) / (cid:15) i ( t ) and g ij ( t ) independently controleach of the ( n + n ) / H n and can therefore be used to simulate any arbitrary, realHamiltonian in H n .While we can simulate H s in H n by choosing (cid:15) i ( t ) and g ij ( t ) so that H n ( t ) = H s ( t ) for all t , a direct mappingbetween Hamiltonians limits the computer to simulatingother quantum systems with similar energy scales overlengths of time within the computer’s coherence time.Fortunately, simulation of H s only requires equality upto an overall phase between U n and the time evolutionoperator generated by H s : U ( t ) ≡ T e − i (cid:126) (cid:82) tt i H s ( t (cid:48) ) dt (cid:48) = e iφ ( t ) U n ( t qc ( t )) . (5)The time elapsed on the quantum computer, t qc ( t ), is astrictly increasing function of simulated time t , admit-ting a much less restrictive relationship between Hamil-tonians: H s ( t ) + c ( t ) = λ ( t ) H n ( t qc ( t )) . (6) c ( t ) is a time-dependent, additive energy shift giving theoverall phase difference φ ( t ) = (cid:126) (cid:82) tt i c ( t (cid:48) ) dt (cid:48) , and we haveintroduced a positive, time-dependent energy/time scal-ing λ ( t ) ≡ dt qc /dt. (7) The energy/time scaling λ ( t ) determines the speed ofthe simulation. By carefully minimizing λ ( t ), we reducethe total simulation time and, consequently, the errordue to decoherence. λ ( t ) is bounded from below by ex-perimental constraints on the allowed values of controlparameters (cid:15) i ( t ) and g ij ( t ) as well as their maximumrates of change. Suppose qubit interaction strengths canvary in a range g ij ( t ) ∈ [ − g max , g max ], and the uncoupledqubit energies can vary in a range (cid:15) i ( t ) ∈ [ (cid:15) min , (cid:15) max ]. Forconvenience, we define a simulated energy E i ( t ) anal-ogous to (cid:15) i ( t ) when diagonal contributions from qubitinteractions are anticipated: E i ( t ) ≡ H ii s ( t ) + α (cid:88) j (cid:54) = i H ij s ( t ) . (8)Using this definition together with equations (3) and (6),we relate the control parameters of the quantum com-puter to the simulated energies in H s ( t ): g ij ( t ) = H ij s ( t ) /λ ( t ) (cid:15) i ( t ) = [ E i ( t ) − c ( t )] /λ ( t ) . (9)By choosing c ( t ) = E max ( t ) − λ ( t ) (cid:15) max where E max ( t ) isthe largest value obtained by the E j ( t ) at a particular t ,we force each (cid:15) i to be as large as possible and thereforeminimize leakage out of H n .Each of the computer’s control parameters remainswithin its allowed range when λ ( t ) is larger than ( n + n ) / λ ( t ) ≥ (cid:40) | H ij s ( t ) | /g max , i (cid:54) = j ∆ E i ( t ) / ∆ (cid:15) max (10)where ∆ E i ( t ) ≡ E max ( t ) − E i ( t ) and ∆ (cid:15) ≡ (cid:15) max − (cid:15) min . λ ( t ) is also bounded by constraints on the speeds withwhich control parameters can change. Suppose v (cid:15)i ( t qc ) ≡ d(cid:15) i ( t qc ) /dt qc and v gij ( t qc ) ≡ dg ij ( t qc ) /dt qc can never belarger in magnitude than v (cid:15) max and v g max respectively.Then for all t , v g max ≥ λ (cid:12)(cid:12)(cid:12)(cid:12) dH ij s ( t ) dt − H ij s ( t ) λ dλdt (cid:12)(cid:12)(cid:12)(cid:12) (11)(and similarly for v (cid:15) max ). To simulate H s ( t ) in H n , we first choose λ ( t ) as smallas both inequalities (10) and (11) allow, guaranteeing afast simulation within the experimental constraints of thequantum computer. We integrate over λ ( t ) to calculate t qc as a function of t : t qc ( t ) = (cid:90) tt i λ ( t (cid:48) ) dt (cid:48) + t qc ( t i ) . (12)With both λ ( t ) and t qc ( t ) known, we can explicitly mapthe matrix elements of H s to the control parameters ofthe quantum computer: (cid:15) i ( t qc ( t )) = (cid:15) max + ∆ E i ( t ) /λ ( t ) g ij ( t qc ( t )) = H ij s ( t ) /λ ( t ) . (13) To demonstrate our theory in detail , we describe threeJosephson phase qubits simulating a three-channel col-lision between a sodium and a helium atom. For threephase qubits with tunable inductive coupling, H qc ( t ) = (cid:88) i =1 − (cid:15) i ( t )2 σ zi + 12 (cid:88) i (cid:54) = j g ij ( t ) ˆΦ i ⊗ ˆΦ j (14)where ˆΦ i is defined in terms of the matrix elements ϕ jk = (cid:10) j | ˆ ϕ i | k (cid:11) of the local Josephson phase operatorin the computational basis of the i th qubit:ˆΦ i ≡ σ xi + ϕ − ϕ ϕ σ zi + ϕ + ϕ ϕ σ i . (15)Both the (cid:15) i and the ϕ jk depend on Φ x , the externallyapplied flux through the superconducting circuit. Ex-ternal flux bias is quantified by a dimensionless param-eter s i ( t ) = Φ x / Φ ∗ x where Φ ∗ x is the qubit’s critical fluxbias, or alternatively, by the dimensionless well depth∆ U/ (cid:126) ω p [14]. We consider external bias values for which s ∈ [ . , .
90] and ∆ U/ (cid:126) ω p ∈ [13 . , . (cid:15)/h = 190MHz while ˆΦ i (cid:39) σ i + 11 σ i varies little. Atunable mutual inductance independently controls thecouplings g ij ( t ) between each pair of qubits. We haveassumed Josephson junction parameters I = 2 . µ A, C = 1 .
52 pF, and L = 808 pH.An n -dimensional subspace can simulate a molecularcollision only after we project the full, many-body Hamil-tonian of the interacting electrons and nuclei into an n -dimensional basis. We construct the collision Hamil-tonian from Born-Oppenheimer energies and nonadia-batic couplings calculated previously for three molecu-lar channels: Na(3 s ) + He(1 s ) [1 Σ + ] and Na(3 p ) +He(1 s )[1 Π; 2 Σ + ] [15], labeled as | (cid:11) s , | (cid:11) s and | (cid:11) s respectively. The energies are stored for fixed valuesof the internuclear distance R , which we assume takesstraight-line trajectories in a standard semiclassical ap-proximation: R ( t ) = √ b + v t where v is the incomingparticle’s velocity and b is the impact parameter of thecollision.Figure 1 outlines our simulation protocol for H s ( t ) de-scribing a three-channel Na-He collision. The matrix el-ements of H s ( t ) are displayed in Fig. 1(a) for a givensemiclassical trajectory R ( t ). Directly below, we plotthe energy/time scaling parameter λ ( t ) as a black curveenveloping the six energy ratios given in Eq. (10). Asmall λ ( t ) speeds the quantum computer through timeswhen the internuclear distance R is large, but as R de-creases ( t → g max value constrainsthe growing couplings. λ ( t ) increases over two orders ofmagnitude, creating a highly nonlinear relationship be-tween t qc and t , as shown in Fig. 1(c). This effectivelystretches the portion of the collision when internucleardistance is small over the entire simulation, as can beseen in the plot of the quantum computer’s control pa-rameters as a function of t qc in Fig. 1(d). FIG. 1: (color online) H s ( t ) describes a three chan-nel Na-He collision with b = 0 . v = 1 .
0. (a)Matrix elements of H s as a function of time in atomicunits (E h = 27 .
21 eV and the atomic unit of time is2 . × − ns). (b) The dimensionless time scalingparameter λ ( t ) envelopes six energy ratios (∆ E = 0 forall t ). We assume g max /h = 2 . (cid:15) max /h =190 MHz. (c) Plot of t qc ( t ) for the case of t qc ( t i ) = 0, t i = −
40 a . u . . (d) Control parameters that simulate H s ( t ) plotted as a function of t qc ( (cid:15) = (cid:15) max for all t qc ). To study the fidelity of the simulation , we compare theexact and simulated time evolution operators, U ( t ) and U n ( t qc ( t )) respectively, by plotting (in Fig. (2)) transi-tion probabilities out of | (cid:11) : P i ( t ) ≡ | (cid:10) i | U ( t ) | (cid:11) | . (16)Because the exact transition probabilities evolve differ-ently with t than the simulated evolve with t qc , we de-fine a time-dependent transition fidelity which accountsfor time scaling, F ( t ) ≡ | s (cid:10) | U † ( t ) U n ( t qc ( t )) | (cid:11) n | , (17)and a time-dependent leakage out of H n , L ( t ) ≡ (cid:80) ⊥ | ⊥ (cid:10) i | U qc ( t qc ( t )) | (cid:11) n | (18)FIG. 2: (a) (color online) Exact transition probabilitiesgenerated by H s ( t ) shown in Fig. 1(a). (b) Transitionprobabilities simulated with parameter profiles given inFig. 1(d). Final simulation fidelity is 0.998.where (cid:80) ⊥ is the sum over all computational basis states | i (cid:11) ⊥ orthogonal to H n . In the upper part of Fig. (3),fidelity and leakage are plotted together for four different g max values.Minimizing g max || J µν || /(cid:15) min , either by decreasing g max or by increasing (cid:15) min , reduces leakage and thus improvessimulation fidelity. In this example, we find simulationfidelity more sensitive to the cutoff in g max because leak-age is most prominent when the interatomic distances aresmall ( t →
0) and the diabatic couplings between chan-nels are the dominant terms. By reducing g max we alsoincrease λ ( t ) and thus the total simulation time, as stud-ied in the lower plot of Fig. (3). To increase fidelity from.9990 to .9999 we need to increase the simulation timeby a factor of ∼
3, a relationship that is independentof n . While not introducing specific models of decoher-ence, we note that high fidelity simulations are possibleon superconducting qubits with coherence times around100 ns.When applied to molecular collisions, our approach toquantum simulation requires classical overhead to projectthe fundamental, time-independent, many-body Hamil-tonian into an R -dependent, n -channel H s . The quanti-ties of physical interest, cross sections, are obtained byintegrating the final transition probabilities over manysemiclassical trajectories with different impact parame-ters, which requires no further classical overhead. A clas-sical simulation of transition probabilities requires ∼ n elementary operations per time step for a single impactparameter, thus cross section calculations are computa-tionally intensive for large n . Alternatively, simulationtime is independent of n using our protocol, so once the R -dependent H s has been calculated, cross sections canbe obtained quickly.In summary, we have presented a straightforward pro-tocol for quantum simulation that can be implementedwith currently available superconducting quantum com-puting technology. While a promising application ofquantum computation, current quantum simulation pro-tocols require a threshold number of gates and qubits FIG. 3: (color online) (a) Fidelity and leakage as afunction of simulation time for four different g max val-ues, with all other parameters the same as in Fig. 1.(b) Final simulation fidelity versus total simulationtime for varying g max . The g max value is referenced bythe shade of the data point.that prohibits fully digital quantum simulations from be-ing demonstrated on available quantum computers. How-ever, we have shown how quantum computers of only afew qubits can simulate arbitrary quantum systems ac-curately and quickly even before they reach the regimeof fault tolerant quantum computation.It is a pleasure to thank Joydip Ghosh for interest-ing discussions. This work was partially supported byNSF grants PHYS-0939849 and PHYS-0939853 from thePhysics at the Information Frontier Program. [1] R.P. Feynman, Int. J. Theor. Phys. , 467 (1982).[2] S. Lloyd, Science , 5273 (1996).; J. L. Dodd, M. A.Nielsen, M. J. Bremner, and R. T. Thew, Phys. Rev. A. , 313-322 (1998).; B. Boghosisanand W. Taylor, quant-ph/9701019 (1997).; D.S. Abramsand S. Lloyd, Phys. Rev. Lett. , 2586 (1997).; L.-A.Wu, M. S. Byrd, and D. A. Lidar, Phys. Rev. Lett. ,057904 (2002).[4] K. R. Brown, R. J. Clark, and I. L. Chuang, Phys. Rev.Lett. , 050504 (2006).[5] A. T. Sornborger and E. D. Stewart, Phys. Rev. A ,1956 (1999).[6] I. Kassal, et al. , PNAS , 18681 (2008); A. Aspuru-Guzik, et al. , Science , 1704 (2005).[7] B. P. Lanyon, et al. , Nature Chemistry , 106 (2010).[8] J. Du, et al. , Phys. Rev. Lett. , 030502 (2010).[9] M. Greiner, et al. , Nature , 39 (2002).; S. Trotzky, etal. Science , 295 (2008).[10] D. Leibfried, et al. , Phys. Rev. Lett. , 247901 (2002);A. Friedenauer, et al. , Nat. Phys. , 757 (2008).[11] S. Somaroo, C. H. Tseng, T. F. Havel, R. Laflamme, andD. G. Cory, Phys. Rev. Lett. , 5381 (1999).[12] I. Buluta and F. Nori, Science , 108 (2009). [13] A. Barenco, et al. , Phys. Rev. A , 3457 (1995).[14] R. A. Pinto, et al. , arXiv:1006.3351; J. Clarke and F.K.Wilhelm, Nature , 1031 (2008). [15] C. Y. Lin, P. C. Stancil, H. P. Liebermann, P. Funke,and R. J. Buenker, Phys. Rev. A78