The Role of Single Qubit Decoherence Time in Adiabatic Quantum Computation

a r X i v :0803.1196v 1 [c o n d -m a t .m e s -h a l l ] 10 M a r 2008

The Role of Single Qubit Decoherence Time in Adiabatic Quantum Computation

M.H.S.Amin,1C.J.S.Truncik,1and D.V.Averin 2

1D-Wave Systems Inc.,100-4401Still Creek Drive,Burnaby,B.C.,V5C 6G9,Canada 2

Department of Physics and Astronomy,SUNY Stony Brook,Stony Brook NY,USA

We numerically study the evolution of an adiabatic quantum computer in the presence of a Markovian ohmic environment.We consider Ising spin glass systems with up to 20coupled qubits that are independently coupled to the environment via two conjugate degrees of freedom.We demonstrate that the required computation time in the presence of the environment is of the same

order as that for an isolated system,and is not limited by the single qubit decoherence time T ?

2,even when the minimum gap is much smaller than temperature.We also show that the behavior of the system can be e?ciently described by a two-state model with only longitudinal coupling to the environment.

Adiabatic quantum computation [1](AQC)is an at-tractive model of quantum computation (QC)as it natu-rally possesses some degree of fault tolerance.In AQC,a system starts from a readily accessible ground state of an initial Hamiltonian H i and slowly evolves into the ground state of a ?nal Hamiltonian H f which encodes a solution to the problem of interest:

H S (t )=[1?s (t )]H i +s (t )H f ,

(1)

where s (t )∈[0,1]is a monotonic function of time t .Here,we only consider linear interpolation for which s (t )=t/t f ,where t f is the total evolution time.Transitions out of the ground state can be caused by a Landau-Zener process [2]at the anticrossing (s =s ?)where the gap g between the ground state |0 and ?rst excited state |1 goes through a minimum:g m ≡g (s ?).The probability of being in the ground state at the end of the adiabatic evolution is approximately (ˉh =k B =1)

P 0f =1?e ?t f /t a ,t a ≡4

ds |0 s =s

?,(2)To ensure a large P 0f one needs t f >～t a .The computa-tion time is hence determined by t a and thus by g m .

In gate model QC there is no direct correspondence between the wavefunction and the instantaneous system Hamiltonian.The Hamiltonian is only applied at the time of gate operations and usually a?ects only a few qubits.The wavefunction,therefore,is strongly a?ected by the environment and is irreversibly altered after the decoherence time,which is typically of the order of the

single qubit dephasing time T ?

2.Thus decoherence im-poses an upper limit on the total computation time,un-less quantum error correction schemes (which require sig-ni?cant resources)are utilized.This is not true for AQC as the wavefunction is always very close to the instanta-neous ground state of the system Hamiltonian and is con-sequently more stable against decoherence.Intuitively,one expects decoherence to drive the density matrix to-wards being diagonal in the energy basis,which is not harmful for AQC but is detrimental for gate model QC.Despite several papers that have demonstrated such ro-bustness [4,5,6,7,8,9],this issue is still a subject

of debate.Much of the criticism stems from the fact

that previous AQC studies have either used a two-state model to describe the behavior of a multi-level system at the anticrossing,or assumed noise models that are not motivated by physical implementations.In this paper,we numerically study the quantum evolution of a multi-qubit system with a quite general and realistic coupling to the environment.

Consider a very general Hamiltonian H (t )=H S (t )+H B +H int ,which includes system,bath,and interac-tion terms,respectively.The dynamics of the total (sys-tem +environment)density matrix are governed by the Liouville equation [10]:˙ρ(t )=?i [H (t ),ρ(t )].The re-duced density matrix for the system is obtained by par-tially tracing over the environmental degrees of free-dom:ρS =Tr B [ρ].Let |n (t ) denote the instantaneous eigenstates of the system Hamiltonian:H S (t )|n (t ) =E n (t )|n (t ) .In such a basis,we de?ne ρnm (t )= n (t )|ρS (t )|m (t ) .Taking the time derivative,we obtain (dropping explicit time dependences)

˙ρnm = n |˙ρS |m + ˙n |ρS |m + n |ρS |˙m .

(3)

Let us begin by focusing on the ?rst term in (3),which is responsible for the decay processes.We treat this term quasi-statically,which means we as-sume that the evolution of the Hamiltonian is much slower than the environmentally induced decay rates so that we can treat the eigenstates as time indepen-dent.We work in the interaction picture in which the density matrix,ρI (t )=U ?(t )ρ(t )U (t ),evolves according to ˙ρI (t )=?i [H I (t ),ρI (t )],with U (t )=e ?i

t

(H S +H B

)dτ

and H I (t )=U ?(t )H int (t )U (t ).Inte-grating the above di?erential equation,we obtain ρI (t )=

ρI (0)?i t

0dτ[H I (τ),ρI (τ)].After one iteration and tak-ing derivatives with respect to t ,we ?nd

˙ρI (t )=?i [H I (t ),ρI (0)]? t

dτ[H I (t ),[H I (τ),ρI (τ)]].(4)

We now introduce a few simplifying approximations.First,we assume that the e?ect of the system on the environment is so small that the bath’s density matrix in

2

the right hand side of (4)can be represented by ρB

(0)

at

all times and that the total density matrix can be written as a direct product:ρI (t )=ρSI (t )ρB (0).We also assume that the bath has a correlation time τB shorter than all decay times so that we can use the Markovian approximation to replace ρSI (τ)with ρSI (t )inside the integral.Tracing over the bath’s degrees of freedom and assuming that the ?rst-order term in H I vanishes after averaging,we ?nd ˙ρSI (t )=?

t

dτTr B [H I (t ),[H I (τ),ρSI (t )ρB (0)]].(5)

After some manipulation,the ?rst term in (3)becomes

n |˙ρS |m =?iωnm ρnm +e ?iωnm t n |˙ρSI |m

=?iωnm ρnm ?R nmkl ρkl ,(6)

where ωnm =E n ?E m ,

R nmkl =δlm Γ(+)

nrrk +δnk Γ(?)

lrrm ?Γ(+)

lmnk ?Γ(?)

lmnk ,

Γ(+)lmnk = ∞

0dt e ?iωnk t ?H

I,lm (t )?H I,nk (0) ,Γ(?)lmnk = ∞

dt e ?iωlm t ?H I,lm (0)?H I,nk (t ) ,(7)

?H

I,nm (t )= n |e iH B t H int (t )e ?iH B t |m .Here, ... ≡Tr B [ρB ...],and summation over repeated indices is implicit.Substituting (6)into (3),we obtain

˙ρnm =?iωnm ρnm ?(R nmkl ?M nmkl )ρkl ,

(8)

where M nmkl =δnk l |˙m +δml ˙n |k .The tensors M nmkl

and R nmkl are responsible for non-adiabatic and thermal transitions,respectively.For a time independent Hamil-tonian,M nmkl =0and (8)becomes the Bloch-Red?eld equations [10,11].

The state derivatives,|˙n etc.,can be calculated nu-merically.It is important to ensure that the equation stays trace preserving,which requires Re n,m n |˙m =0.This condition is exactly satis?ed (even with the truncation discussed below),if we write n (t )|˙m (t ) =

1

2

i,α

S (i )

α(?ωnk )σ(i )α,lm σ(i )

α,nk ,

Γ(?)

lmnk =

1

E =?

1E

=?1

2

i>j

J ij σ(i )z σ(j )

z ,

(12)

where σ(i )

x,z are the Pauli matrices corresponding to the

i -th qubit,E is an energy scale,and ?i ,h i ,and J ij are dimensionless parameters.We consider square lat-tice con?gurations with nearest and next-nearest neigh-bor coupling between the qubits.We generate spin glass instances involving 6,9,12,16,and 20qubits by ran-domly choosing h i and J ij from {?1,0,1}and identify-ing small gap instances with non-degenerate ?nal ground state (see,e.g.,Fig.1).Such instances are very rare and represent di?cult problems;a degenerate ground state (multiple solutions)ensures higher probability of ?nding one of the solutions.We also choose ?i =1for all i .Figure 1shows the energy spectrum for a typical 20qubit instance with a small g m .The ?rst two energy lev-els anticross near the middle of the evolution.In the same

3

t f (m s)

P 0

f

FIG.2:Probability of success P 0f as a function of t f for the 20qubit instance of Fig.1.The solid lines are calculated with (η=0.2)and without (η=0)coupling to the environment.Other parameters are E =10GHz,T =25mK.The dashed lines are obtained using analytical formulas (2)and (15).The dotted line is numerical calculation with η=0.2θ(T ?g ).

?gure,we have also displayed the ground state entangle-ment calculated using a measure originally proposed by Meyer and Wallach [13]:

Q (|ψ )=

1

n

i

|σ(i )

α,10|2)1/2,which give the r.m.s.values

of the matrix elements of Pauli matrices σ(i )

αbetween the lowest two states.They represent some average be-havior of the corresponding matrix elements.Moreover,

4

FIG.3:(a)Matrix elements as a function of s .Solid (blue)line is M z ,dashed-dotted (black)line is M x and the dashed

(red)line is M TSM

z obtained from a two state model keeping only the longitudinal coupling to the environment.The inset shows the same curves zoomed near the anticrossing.

if all qubits have the same coupling ηto the environ-ment,then the relaxation rate between the two states

is given by γ=n (M 2x +M 2z )?S (ω10),where ?S

(ω)is the symmetrized

spectral

density

of the (uncorrelated)baths.Figure 3(a)displays M x and M z as a function of s for the system of Fig.1.Except for the initial region,they both show the same behavior:a sharp peak at the anti-crossing,with a width proportional to g m ,followed by a vanishingly small value.For small T the excitation from the ground state will be suppressed everywhere except near the anticrossing where g

The transitions at the anticrossing can be e?ciently described by an e?ective two-state Hamiltonian:

H TSM

S =?

12

(1?e ?2t f /t a ),

(15)

This formula is also plotted in Fig.2.The qualitative agreement with other curves on the ?gure indicates that most of the transition occur in small gap regions (g ?T )and TSM is adequate to describe such processes.To summarize,by studying spin glass instances of up to 20qubits (only one is illustrated),we have explicitly demonstrated that the computation time in AQC can be

much longer than single qubit decoherence time T ?

2.We have also shown that a two-state model with longitudi-nal coupling to the environment can e?ciently describe the physics of transitions at the https://www.sodocs.net/doc/4015259160.html,ing such a model,the computation time scale was shown to be una?ected by an ohmic environment,in agreement with the numerical results.This e?ect was not due to suppres-sion of transitions by a gap,as the minimum gap in the instances we have chosen was much smaller than tem-perature and decoherence strength.It should be empha-sized that the above results were obtained under the as-sumption of weak coupling to the environment,for which the discrete energy structure of H S is approximately pre-served (except for some renormalization).For strong cou-pling to the environment,the interaction Hamiltonian will dominate and the above method will not hold.The authors are grateful to A.J.Berkley,P.Bunyk,V.Choi,R.Harris,J.Johansson,M.W.Johnson,S.Lloyd,and G.Rose for useful discussions.

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