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Y.WANG
yields M. The combination of the quantum count- quantum systems for which efficient simulation by
ing and search procedure will find a solution of the classical computers may not be available.
search problem with certain probability. Repeating
5.1 Simulate a Quantum System
the quantum search algorithm will boost the prob-
ability and enable us to obtain a solution to the The key of quantum simulation is to solve the
search problem. Schro¨dinger equation (1) which has solution
Quantum walk and quantum Markov chain are (14) ψ(t) =e iHt ψ(0) , i=√ 1.
−
currentlybeinginvestigated fornewquantumsearch | i | i −
−iHt
algorithms and quantum speed-up of Markov chain Numericalevaluationofe isneeded.TheHamil-
based algorithms (Aharonov and Ta-Shma (2003), tonianHisusuallyexponentiallylargeandextremely
Childs et al. (2003); Childs (2010); Tulsi (2008); difficult to exponentiate. The common approach in
Shenvi, Kempe and Whaley (2003) and Szegedy, numerical analysis is to usethe first-order linear ap-
−iH(t+δ) −iHt,
2004).InSection5weshowthatthequantumsearch proximation, 1 iHδ, of e e which
− −
algorithm can also be viewed as a quantum simula- often yields unsatisfactory numerical solutions.
tion procedure. Many classes of Hamiltonians have sparse repre-
sentations.ForsuchsparseHamiltonianswecanfind
5. QUANTUM SIMULATION efficient evaluation of the solutions (14) with high-
order approximation. For example, the Hamiltoni-
Quantum simulation is to intentionally and artifi-
ans in many physical systems involve only local in-
cially mimic a natural quantum dynamics, which is
teractions, which originate from the fact that most
hard to access, and analyze, by a computer-gener-
interactions fall off with increasing distance in lo-
ated quantum system, which is easy to manipulate
cation or increasing difference in energy. In the lo-
and investigate. It provides scientific means for sim-
calHamiltoniancase,theHamiltonianofaquantum
ulating complex biological, chemical or physicalsys-
systemwithαparticles inad-dimensionalspacehas
temsinordertostudyandunderstandcertainscien-
the form
tific phenomena and evaluate hard-to-obtain quan-
tities in the systems. Examples in modern scien- L
(15) H=2 H ,
tific studies include the estimation of dielectric con- ℓ
Xℓ=1
stant, proton mass, and precise energy of molecu-
lar hydrogen, the study of superconductivity, the where L is a polynomial in α+d, and each H acts
ℓ
test of novel nano-materials, and the design of new on a small subsystem of size free from α and d. For
biomolecules. example, the terms H are typically two-body inter-
ℓ
To simulate a quantum system we need to solve actions and one-body Hamiltonians. Hence e −iH ℓδ
iHδ
the Schro¨dinger equation (1) which governs the dy- areeasytoapproximate,althoughe isveryhard
−
namic evolution of the system. For a typical Hamil- to evaluate. Since H and H are non-commuting,
ℓ k
tonian with real particles the Schro¨dinger equation e iHδ =e iH 1δ e iH Lδ. Applying a modification
− − −
6 ···
usually consists of elliptical differential equations, of the Trotter formula (Kato (1978); Trotter (1959);
each of which can be easily simulated by a classical Yu (2001)) we obtain
computer. However, the real challenge in simulating e iHδ =U +O(δ2),
a quantum system is to solve the exponential num- − δ
ber of such differential equations. For a quantum where
system of b qubits, its states have 2b amplitudes. To (16) U =[e iH 1δ e iH Lδ][e iH Lδ e iH 1δ].
δ − − − −
simulate the dynamic behavior of b qubits evolving ··· ···
−iHδ
according to the Schro¨dinger equation, we need to Thus we can approximate e by U δ which needs
solve a system of 2b differential equations. Due to to evaluate only each e −iH ℓδ.
the exponential growth in the number of differential Assume that the quantum system starts at t=0
equations, the simulation of general quantum sys- with initial state ψ(0) and ends at final time t=1.
| i
tems by classical computers is very inefficient. Clas- For an integer m, set t j =j/m, j=0,1,...,m. The
sical simulation of quantum systems is feasible for quantumsimulationistoapplyapproximationU δ of
−2iHδ
the cases where insightful approximations are avail- e toevaluate(14)att j iteratively andgenerate
able to dramatically reduce the effective number of approximate solutions to ψ(t ) . Denote by ψ˜(t )
j j
| i | i
equations involved. Quantum computers may excel the state at t obtained from the quantum simula-
j
in simulating physically interesting and important tion as an approximation of the true state ψ(t )
j
| i
17
QUANTUMCOMPUTATIONAND QUANTUMINFORMATION
at t . Then for j=1,...,m, Carlotechniquesinthesimulationsofsuchquantum
j
2iHδ 2iHjδ systems. The combination of Monte Carlo methods
ψ(t ) =e ψ(t ) =e ψ(t ) ,
(17) | j i − | j −1 i − | 0 i with quantum simulation makes it possible to ob-
ψ˜(t ) =U ψ˜(t ) =Uj ψ(t ) . tain reliable quantifications of quantum phenomena
| j i δ | j −1 i δ| 0 i and estimates of quantum quantities. Such combi-
While classical computers are inefficient in simu-
nation procedures are often referred to as quantum
lating general quantum systems, quantum comput-
Monte Carlo simulation (Nightingale and Umrigar
ers can efficiently carry out the quantum simulation
(1999); Rousseau (2008)). Consider the problem of