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procedure and provide an exponential speedup for
estimating the following quantity:
the quantum simulation on classical computers. In
(18) θ=Tr(Xρ)=E(X),
spite of the inefficiency, classical computers are cur-
rently being used to simulate quantum systems in where X is an observable, X is its measurement re-
biochemistry and material science. Quantum simu- sult,andρisthestateofthequantumsystemunder
lation will be among the important applications of which we perform the measurements and evaluate
quantum computers. See Abrams and Lloyd (1997), the quantity θ.
Aspuru-Guzik,DutoiandHead-Gordon(2005),Ben- Asρisthetruefinalstateofthequantumsystem,
nettetal.(2002),Berryetal.(2007),Freedman,Ki- we denote by ρ˜ the final state of the quantum sys-
taevandWang(2002),Jan´eetal.(2003),Boghosian temobtainedviaquantumsimulation.Thequantum
andTaylor(1998),Lloyd(1996),NielsenandChuang systems are prepared in initial state ρ , and we use
0
(2000), and Zalka (1998).
the quantum simulation procedure described above
5.2 Recast Quantum Search Algorithm as tosimulatetheevolutionsofthesystemsfrominitial
Quantum Simulation state ρ 0 to final state ρ˜ according to Schro¨dinger’s
equation (14) with Hamiltonian H given by (15).
Grover’s search algorithm discussed in Section 4.5
We repeatedly perform the measurements of such n
is an important finding in quantum computation. It
identically simulated quantum systems at the final
can be heuristically sketched as a quantum simu-
state and obtain measurement results X ,...,X .
1 n
lation by writing down an explicit Hamiltonian H
We estimate θ defined in (18) by
such that a quantum system evolves from its initial
n
state ψ to x after some specified time, where x 1
| i | i (19) θˆ= X .
is a solution of the search problem. Of course the n j
Xj=1
Hamiltonian H depends on the initial state ψ and
| i
solution x. Suppose that y is another state such The target θ given by (18) is defined under the
| i
that x and y form an computational basis, and truestate ρ,while the simulated quantum system is
| i | i
ψ =αx +β y for real α and β with α2+β2=1. under approximate final state ρ˜ which is close to ρ.
| i | i | i
Define Hamiltonian The measurement results X ,...,X are obtained
1 n
via quantum simulation from the quantum systems
H= x x + ψ ψ =I+α(βσ +ασ ),
x z
| ih | | ih | inthesimulatedstateρ˜.Therefore,theMonteCarlo
where σ x and σ z are Pauli matrices defined in (7). quantum estimator θˆin (19) involves both bias and
Then variance. Wang (2011) studied the quantum simu-
exp( iHt)ψ lation procedure and investigated the bias and vari-
− | i ance of θˆ. The derived bias and variance results can
=e it[cos(αt)ψ isin(αt)(βσ +ασ )ψ ]
− x z be used to design optimal strategy for the best uti-
| i− | i
=e it[cos(αt)ψ isin(αt)x ]. lization of computational resources to obtain the
−
| i− | i
quantum Monte Carlo estimator.
Measuring the system at time t=π/(2α) yields the
solution state x .
6. QUANTUM INFORMATION
| i
5.3 Quantum Monte Carlo Simulation
Classical information theory is centered on Shan-
Quantum theory is intrinsically stochastic and non’s two coding theorems on noiseless and noisy
quantummeasurementoutcomeisrandom.Asmany channels.Thenoiselesschannelcodingtheoremquan-
naturallyoccurringquantumsystemsinvolvealarge tifies the number of classical bits required to store
number of interacting particles, due to the compu- information for transmission by Shannon entropy,
tational complexity we are forced to utilize Monte while the noisy channel coding theorem quantifies
18
Y.WANG
theamountofinformationthatcanbereliablytrans- and intriguing structure of entangled states and re-
mittedthroughanoisychannelbyanerror-correction markable connections between noisy quantum chan-
coding scheme. The quantum analogs of Shannon nels and entanglement transformation. Consider
entropy and Shannon noiseless coding theorem are quantum error-correction for reliable quantum com-
von Neumann entropy and Schumacher’s noiseless putationandquantuminformationprocessing.Quan-
channel coding theorem, respectively. The von Neu- tum error-correction is employed in quantum com-
mann entropy is defined to be S(ρ)= tr(ρlogρ). putation and quantum communication to protect
−
Schumacher’snoiselesschannelcodingtheoremquan- quantuminformationfromlossduetoquantumnoise
tifies quantum resources requiredto compressquan- and other errors like faulty quantum gates. Classi-
tum states by von Neumann entropy (Schumacher cal information uses redundancy to achieve error-
(1995)). Analogous to Shannon’s noisy channel cod- correction, but the no-cloning theorem presents an
ing theorem, a theorem known as Holevo–Schuma- obstacle to copying quantum information and for-
cher–Westmoreland theorem can be used to com- mulatingatheoryofquantumerror-correctionbased
pute the product quantum state capacity for some onsimpleredundancy.Againquantumentanglement
noisychannels(Holevo(1998);SchumacherandWest- comes to the rescue. It is forbidden to copy qubits
moreland (1997)). However, communications over butwe can spread theinformation of onequbitonto
noisy quantum channels are much less understood ahighlyentangledstateofseveralqubits.Shor(1995)
thantheclassicalcounterpart.Itisanunsolvedprob- first discovered the method of formulating a quan-
lem to determine quantum channel capacity or the tumerror-correctioncodebystoringtheinformation
amount of quantum information that can be reli- of one qubit onto a highly entangled state of nine
ably transmitted over noisy quantum channels. See qubits. Over time several quantum error-correction
Hayashi (2006) and Nielsen and Chuang (2000). codes are proposed (Calderbank and Shor (1996);
In spite of the above similarity, there are intrinsic Cory et al. (1998); Steane, 1996a, 1996b). These
differences between classical information and quan- quantum error-correction codes can protect quan-
tum information. Classical information can be dis- tum information against quantum noise, and thus
tinguished and copied. For example, we can identify quantum noise likely poses no fundamental barrier
different letters and produce an identical version of to the performance of large-scale quantum comput-