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Encyclopedia of Algorithms
DOI 10.1007/978-3-642-27848-8_317-2
© Springer Science+Business Media New York 2015
Quantum Search
Lov K. Grovera and Apoorva D. Patelb
a
Bell Labs, Alcatel-Lucent, Murray Hill, NJ, USA
b
Centre for High Energy Physics, Indian Institute of Science, Bangalore, India
Keywords Amplitude amplification • Grover diffusion • Quantum walk • Unsorted database •
Unstructured search
Years and Authors of Summarized Original Work
1996; Grover
Problem Definition
Brief Description
The search problem can be described informally as finding an item possessing a specific property,
in a given set of N items. Each item either does or does not possess the specified property, and
that can be checked by a binary query. The complexity of the problem is the number of such
queries required to find the desired item (also called the target item). The items are often collected
in a database and sorted to simplify the subsequent searches. When they are not sorted, there is
no shortcut to the brute force method of checking each item one by one until the desired item is
found. A familiar example of a database is a telephone directory. Its entries are sorted according
to the names of persons but not according to the telephone numbers. Hence, it is easy to find the
telephone number of a particular person, but difficult to find the name of a person to whom a
particular telephone number belongs (i.e., a lookup is difficult when it is not in the same order in
which the database is sorted).
The .N / lower bound on search speed, based on inspection of one item at a time, is correct
only for classical computers. Quantum computers can be in a superposition of multiple states,
however, and so can inspect multiple items at the same time. There is no obvious lower bound on
how fast a quantum search can be, nor is there an obvious technique faster than the brute force
search. It turns out,
p though, that there is an efficient and optimal quantum search algorithm that
requires only O. N / queries [15].
This quantum algorithm is very different from the search on a classical computer [8]. The
optimal classical strategy is to check the items one at a time in a random order, avoiding in later
trials the items that have already been checked earlier. After ˜ items have been checked from
a uniform distribution, the probability that the search hasn’t yet succeeded is .1 1=N /.1 1=.N 1// .1 1=.N ˜ C 1//. For N , the success probability is therefore roughly
1 .1 1=N /˜ ˜=N . Increasing this success probability to ‚.1/ requires the number of items
checked, ˜, to be ‚.N /.
E-mail: [email protected]
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In contrast to classical computation, quantum computation is formulated in terms of wavelike
complex amplitudes, whose interference can be used to cancel undesirable components and boost
the desired component. Quantum search is then analogous to the design of a multi-element antenna
array, where a careful choice of phases can boost the radiation in a particular direction. The
analysis of such structures is carried out using the algebra of unitary transformations, and absolutevalue squares of the amplitudes give the observation probabilities. Unitary transformations include
rotations and reflections (about various directions) in the space of amplitudes, as well as local
phase shifts. Rotations and reflections redistribute the amplitudes and are similar to classical
transformations. On the other hand, the phase shifts are uniquely quantum; they do not alter
probabilities of individual components, but affect their subsequent interference pattern. The
challenge of quantum computation is to find a sequence of elementary unitary operations (i.e.,
quantum logic gates) that solve the given computer science problem, while ensuring that the input
and the output of the quantum algorithm have clear classical interpretations.
The quantum search algorithm steadily increases the amplitudepof the desired item through a
series of quantum operations.
p Starting with an initial amplitude 1= .N /, in steps, the amplitude
increases to roughly = N , and hence, the success probability
(on observation of the state)
p
increases to 2 =N . Boosting this to ‚.1/ requires only O. N / steps, approximately the square
root of the number of steps required by the best classical algorithm.
The quantum search algorithm is of wide interest because of its versatility; it can be adapted to
different settings in a variety of fields, giving a new class of quantum algorithms extending well
beyond the search problems. Since its discovery, it has been incorporated in solutions of many
quantum problems – several of them are mentioned later in this article. Even now, two decades
after the algorithm’s discovery, new applications and extensions keep on appearing regularly.
Formal Construction
Let the items in the set be labeled by an index i D 1; 2; : : : ; N . Let the binary query be represented
by an oracle f .i /, such that f .i / D 1 when i represents a desired item and f .i / D 0 otherwise.
The quantum algorithm works in an N -dimensional vector space with complex coordinates, known
as the Hilbert space. We use Dirac’s notation, which is standard in the literature of quantum
mechanics and quantum computation. Then the items are mapped to the N orthogonal basis vectors
ji i of the Hilbert space, and the binary query is mapped to the selective phase-shift operator defined
by Uf ji i D .1/f .i/ ji i. Given the binary query f .i /, it is easy to construct the operator Uf using
an ancilla qubit.
The problem is to start from a specific initial state jsi and evolve to a target state satisfying
Uf jt i D jt i, by applying a sequence of unitary operations. The number of times Uf is used
in the algorithm is its query complexity. This search problem is unstructured because nothing is
known about the solution, except the information available from the oracle that can tell whether or
not a specific state is the target state.
NP-complete problems can be represented as exhaustive search problems. For example, let ¥ be
a 3-SAT formula on n Boolean variables. Then the search problem is to find an assignment for the
variables, i 2 f1; 2; : : : ; N D 2n g, that satisfy ¥. This example does not involve a database and so
bypasses concerns regarding how the items are stored in a physical memory device and the spatial
relationship among them.
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Key Results
Grover [15] showed that there indeed exists a quantum search algorithm that provides a squareroot speedup over the optimal classical randomized algorithm. The algorithm has its simplest
form when there is only one target item. Then the algorithm starts with an unbiased uniform
N
p
p
P
ji i and performs Q D O. N / iterations to evolve to
superposition state jsi D .1= N /
iD1
the state (UD Uf /Q jsi. Each iteration consists of two reflection operations:
1. Uf D 12jt iht j is a reflection along jt i. It uses the binary query to flip the sign of the amplitude
of the target state.
2. UD D 2jsihsj 1 is reflection about jsi. It can be carried out without any information about
the target state. Since the action of jsihsj gives the average amplitude state, UD amounts to
inversion about the average or overrelaxation.
At the end, the final state is measured in the fji ig basis that encodes the item labels, and i is output.
There are several ways of analyzing the algorithm, and the geometric picture is perhaps the
simplest. We observe that throughout the evolution, the quantum state stays in the two-dimensional
subspace (of the p
Hilbert space) spanned by jsi and jt i. Initially, the amplitude of thep
state along
jt i is ht jsi D 1= N , and the angle between jt i and jsi is =2 ™ with sin D 1= N . It is a
general property of linear transformations in two dimensions that a pair of reflections about two
distinct axes produces a rotation, and the amount of rotation is twice the angle between the two
axes. The quantum search algorithm is an alternating sequence of reflections about two different
axes. Each application of the operator UD Uf rotates the quantum state from jsi toward jt i by
angle 2™. The number of iterations QQ required to exactly reach the target state is therefore given
by .2QQ C 1/ D =2. In practice, we have to truncate to integer Q D bQQ C 0:5c, introducing
2
a small error.
p The success probability still remains at least cos ™ D 1 1=N . Asymptotically,
Q D .=4/ N .
The reflection about the uniform superposition state, UD , is known as the Grover diffusion
n
operation. When the indices are represented in binary notation,
with N D 2 , we have jsi D
1 1
H ˝n j0i˝n in terms of the Hadamard operator H D p12
. Then UD D H ˝n U0 H ˝n , with
1 1
U0 D 2j0i˝n h0j˝n 1 being the reflection about the j0i˝n state, and it can be implemented using
O.n/ qubit-level operations. In this case, the full quantum search algorithm evolves the state j0i˝n
to the state (H ˝n U0 H ˝n Uf /Q H ˝n j0i˝n .
When there are M target items, instead of just one, all that is required is to replace jt i by
M
p
P
jtj i= M in the algorithm. The final measurement then yields one of the target items after
j D1
p
O
N=M queries. Thus, we have [15]:
Theorem 1 (Grover search). There is a quantum black-box unstructured search algorithm with
success
‚.1/, which finds any one of the M target items in a set of N items, using
p probability
N=M queries.
O
This algorithm has several noteworthy properties:
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p • The algorithm is optimal. It saturates the N lower bound [5] on the number of queries
required for an unstructured quantum search. The evolution from jsi to jt i follows the shortest
geodesic route in the Hilbert space at constant speed. A variational analysis shows that the
algorithm cannot be improved by even a single query [43].
• The best classical search algorithm has to walk randomly through all the items, while the
quantum search algorithm performs a directed walk in the Hilbert space. The square-root
speedup of the quantum search algorithm can therefore be understood as the well-known result
that directed walk provides a square-root speedup over random walk while covering the same
distance.
• The algorithm can be looked upon as evolution of the quantum state from jsi to jt i, governed
by a Hamiltonian containing two terms, jt iht j and jsihsj. The former represents a potential
energy attracting the state toward jt i, and the latter represents a kinetic energy diffusing the state
throughout the Hilbert space. The algorithm is then the discrete Trotter’s formula, generated by
exponentiating the two terms in the Hamiltonian [17].
• Grover search does not require the full power of quantum dynamics and can be implemented
using any system that obeys the superposition principle. Explicit examples using classical waves
in the form of coupled oscillators have been constructed [20, 29]. In these mechanical systems,
the role of the uniform superposition state is played by the center-of-mass mode, and the search
problem becomes the energy focusing problem. The classical wave implementation requires the
same number of queries as the quantum algorithm. The difference is that to represent N items,
we need N wave modes but only n D log2 N qubits.
• Grover search finds with certainty a single target state out of four possibilities using a single
binary query, i.e., Q D 1 for N D 4. The best classical Boolean algorithm can distinguish
only two items with a single binary query, and so it needs two binary queries to carry out the
same task. When the query can be factored into subqueries, e.g., the item label is searched for
one digit at a time and not as a whole, the best quantum (or wave) arrangement for a database
is a quaternary tree. Then every subquery reduces the search space by a factor of 4, which is a
factor-of-2 advantage over the classically optimal binary tree. An additional advantage following
from commutativity of superposition is that, unlike the classical case, the quantum tree does not
require sorting of the database [28].
• The quantum search algorithm is robust against changes in the initial state and the operators, in
sharp contrast to many other quantum processes that are highly sensitive to errors. The initial
state and the diffusion operator are related in Grover search, but can be separated in a more
general context [2, 40]. Let UQ D be the modified diffusion operator with jsi as an eigenstate with
eigenvalue 1, i.e., the diffusion is translationally invariant. The algorithm then succeeds with
‚.1/ probability, provided ’ D jht jsij as well as the angular spectral gap of UQ D in the vicinity of
identity (say Q ) are bounded away from zero. The number of queries required is O.B 3 =’/, where
B 2 is related to the second moment of the eigenvalue distribution of UQ D and obeys B 2 < 1 C
.4=Q 2 /, in contrast to the classical result O.1=’2 /. Grover search, therefore, can be generalized
to an entire class of algorithms that use different diffusion operators. This flexibility is one of
the reasons why Grover search ideas appear frequently in quantum algorithms.
• Quantum search can be implemented so as to be robust also against faulty queries, a problem
known as bounded-error search. When the query has a bounded coherent error, say UQ f ji ij1i D
p
p
pi .1/f .i/ ji ij1i C 1 pi jiQ ij0i with each pi 0:9 (jiQ i are arbitrary states
pand the last qubit
is a witness for the fault), quantum search can still be implemented using O
N queries [22].
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The deterioration of the search algorithm depends on pi , but that is only in the scaling constant
for the number of queries.
A useful generalization of the quantum search algorithm is the amplitude amplification technique
[10,16], which can be applied on top of nearly any quantum algorithm for any problem. It says that
given a quantum algorithm that solves a problem with a small success probability , the success
probability can be increased to roughly m2 using O.m/ calls to that algorithm. (Classically, the
success probability can be increased to only about m.) For the standard search problem, the simple
algorithm that picks a random item has success probability D 1=N , which the quantum search
algorithm increases to ‚.1/.
More formally, let V be the unitary operator corresponding to an algorithm that evolves the
initial state jsi to V jsi. Its success probability is D jht jV jsij2 D jVt s j2 . The algorithm obtained
by replacing jsi by V jsi in Grover search,
i.e., replacing UD by V UD V , then increases the success
p
probability to ‚.1/ in O 1= jVt s j iterations. In particular, this algorithm evolves the quantum
state in the two-dimensional subspace spanned by jt i and V jsi, rotating it by angle 2 sin1 jVt s j at
every iteration. In order to implement Uf , the algorithm needs a witness for the correctness of the
output. Thus, we have [10, 16]:
Theorem 2 (Amplitude amplification). Let A be a quantum algorithm that outputs a correct
answer with witness, with known probability D sin2 . Furthermore, let m D b =.4™/c. Then
there is an algorithm A0 that uses 2m C 1 calls to A and A1 and outputs a correct answer with
probability 0 1 .
Depending on the actual implementation, it is possible to vary the quantum search algorithm
somewhat from the preceding deterministic and optimal approach and obtain small improvements:
p
N log N qubit-level operations in order to implement H ˝n and
• The algorithm needs O
U0 . The log N factor in this count can be suppressed by adding a small number of queries to
the algorithm. A simple scheme divides the n qubits into k sets of n=k qubits each and uses
the Grover diffusion operators UD .i/ D H ˝.n=k/ U0 .i/ H ˝.n=k/ that act only on one set at a time
leaving the other sets unchanged [18]. Sequentially going through all the sets generates the
k
Q
.i/
.UD Uf /H ˝n , which is then used for amplitude amplification with the
transformation V D
iD1
initial state j0i˝n . Overall, the number of qubit-level operations reduce by a factor ‚.k/, while
1=k
D o.1/. The choice
the number of queries go up by a factor 1 C ‚.kN 1=k /, provided kN
p
k D ‚.n= log n/ reduces the qubit-level operations to O
N log n , at the cost of increasing
the queries by a factor 1 C ‚.1= log n/.
• Consider the partial search problem where the items are separated into N=b blocks of size b
each, and only the block containing the desired item is to
p be located using the same Uf . In
that case, the number of queries can be reduced by 0:34 b for large b [25]. The procedure
first uses Grover search to make the amplitudes of nontarget blocks sufficiently small, then
applies Grover search in parallel within each block to make the amplitudes of the target block
sufficiently negative (amplitudes of nontarget blocks remain unchanged in this step), and then
executes a final UD operation to reduce the amplitudes of nontarget blocks to zero.
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• Though Grover search proceeds from jsi to jt i with uniform speed in the Hilbert space, it slows
down in terms of the success probability as it nears the target state. So one can reduce the
expected number of queries, by stopping the algorithm before reaching the target state and then
looking for the desired item probabilistically.
That amounts
to minimizing .Q C 1/=p, with the
p 2
1
success probability p D sin .2Q C 1/ sin .1= N / . This probabilistic search reduces the
p
required number of queries asymptotically to 0:6900 N [14].
• Consider the search problem where the times required for querying different items are not the
same. This can happen when the query is an algorithm A acting on different input states ji i.
When the query for the i th item takes time!ti to execute, unstructured quantum search can be
N 1=2
P 2
accomplished in time O T D
ti
when ti are known apriori [3]. The strategy is to
iD1
divide the items into multiple groups so that items in every group have query times within a
constant factor, apply Grover search within each group, and then query the groups sequentially.
The number of groups
needed
is O.log N /, and the result improves upon the global Grover
p
N tmax . Amplitude amplification can be used when A is probabilistic, and
search bound O
a polylogarithmic overhead in T is required when ti are not known in advance.
Applications
NP-complete problems: Even though NP-complete problems have some structure, there are few
known algorithms that exploit this structure to solve them, and often the only recourse left is to
solve them as exhaustive search problems. Since quantum search does not assume any structure or
pattern in the input data, it provides a square-root speedup in such cases.
Quantum counting: The counting problem is to find the number of items in a set that satisfy
the given query. Its quantum solution is based on thefact thatthe iterative evolution in Grover
p
M=N . The phase estimation procedure
search is periodic, with angular frequency ! D 2 sin1
(based on quantum
transform) [24] can therefore determine M approximately, up to error
p Fourier
p
p
M , using O
N queries. Then, using the property that ¨ differs by 1= M.N M / between
p
adjacent values of M , M can be determined exactly using O= M.N M / queries [27]. For
M D o.N /, this quantum result is a power-law improvement over the classical result of ‚.N /
queries, although not as good as a square-root speedup.
Element distinctness: An early application of Grover search was to find collisions, i.e., given
oracle access to a 2-to-1 function f , find distinct arguments x; y such that f .x/ D f .y/. The
quantum collision problem has an O.N 1=3 / algorithm [9]. The more general element distinctness
problem is to find distinct x; y such that f .x/ D f .y/, for an unknown function f that can
be accessed only by an oracle. Ambainis discovered an optimal O.N 2=3 / quantum algorithm for
this problem [2]. It searches a suitably constructed graph, with the Grover diffusion operation
replaced by a certain quantum walk. The vertices of the graph correspond to various subsets of
items Sj f1; 2; : : : ; N g, each of size N 2=3 , two vertices are connected by an edge when the
corresponding subsets differ by only one item, and the target vertices are the subsets Sj that solve
the element distinctness problem.
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Distributed search: Grover search is also useful in improving communication complexity. For
example, a straightforward distributed implementation of the quantum search algorithm solves the
set intersection problem or the appointment problem. The result is, when A and B have respective
N
data
pstrings x;y 2 f0; 1g , and they want to find an index i such that xi D yi D 1, only
N log N qubits of communication is necessary [11]. This result has led to an exponential
O
classical/quantum separation in the memory required to evaluate a certain total function with a
streaming input [26].
Fixed-point search: The iterative evolution in Grover’s search algorithm is cyclic, and knowledge
of N is necessary to stop it at the right time to find the target item. In contrast, fixed-point search
algorithms converge monotonically to the target state. For a long time, a fixed-point quantum
algorithm was considered unlikely, since any iterative unitary evolution is periodic. Surprisingly,
a way out was provided by recursive unitary evolution. When the reflection operations in the
amplitude amplification algorithm are replaced by selective phase shifts of =3 (e.g., Ri D
1 C .e i =3 1/ji ihi j for the state ji i), jht jV jsij2 D 1 implies jht jVRs V Rt V jsij2 D 1 3 [19].
So each recursive substitution of the operator V by the operator VRs V Rt V reduces the deviation
of the final state from the target state to the cube of what it was before. (The corresponding best
classical reduction is O. 2 /, e.g., by majority rule selection after three trials.) This technique does
not give a square-root speedup for search, but it is useful when is small, for instance, in error
correction. It has been used to design composite pulse sequences for reducing systematic errors
[35]. An iterative quantum search algorithm with similar properties has been obtained combining
reflection operations with non-unitary projective measurements [41]. Another recent construction
is a bounded-error quantum search algorithm (i.e., success probability p 1 •) that varies the
phase shifts between and =3 as a function of the iteration number [42]. It exhibits square-root
speedup as well as convergence to the target state, provided that both • and jht jsij are bounded
away from zero.
Spatial search: This is the search problem where the items belonging to a database are spread
over distinct physical locations, say a d -dimensional lattice, and there is a restriction that one can
proceed from any location to only its neighbors while searching for the target item. Its quantum
solution replaces the global Grover diffusion operator by a local quantum walk, and Grover search
becomes
the d !
1 limit. The required number of queries has to obey the double lower bound
p 1=d
dN ; N ; the former arises from the finite speed of movement on the lattice and the latter
from the optimality of Grover search.
p The
best algorithms are found in the framework of relativistic
quantum mechanics. They use O
N queries for d > 2, with the scaling constant approaching
=4 from above as d ! 1 [1, 33]. In the critical dimension d D 2, the algorithms are slowed
down by logarithmic
factors
arising from the infrared divergence, and the best known algorithm
p
N log N queries [39]. For non-integer values of d , the scaling behavior of the
requires O
algorithm has been verified using numerical simulations on fractal lattices [32].
Markov chain evolution: Generic stationary stochastic processes (e.g., random walks) are
defined in terms of transition matrices that encode the possible evolutionary changes at each
step. Many properties of the resulting evolution (e.g., hitting time, detection, mixing, escape time)
scale as negative powers of the spectral gap of the transition matrix. For Markovian evolution on
bipartite graphs, the transition matrix can be separated into two disjoint parts, say fxg ! fyg and
fyg ! fxg. Szegedy constructed two reflection operators from these parts and defined a quantum
evolution operator as their product [38] (classical Markov chain evolution does not allow such
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reflection operators). The spectral gap of this quantum evolution operator scales as the square root
of the spectral gap of the original transition matrix and so speeds up the evolution the same way as
Grover search does.
Recursive search: Game-tree evaluation, which is a recursive search problem, is an extension of
unstructured search. Classically, using the alpha-beta pruning technique, thevalue of a balanced
p
binary AND-OR tree can be computed with o(1) error in expected time O N log2 Œ.1C 33/=4 D
O.N 0:754 / [36]. This is optimal even for bounded-error algorithms [37]. By applying quantum
search recursively, a depth-d regular AND-OR tree can be evaluated with constant error in time
p
N O.log N /d 1 . The log factors come from amplifying the success probability of inner searches
to be
pclose toone. Bounded-error quantum search eliminates these log factors, reducing the time to
O
N c d for some constant c. Recently, an O.N 0:5Co.1/ / time algorithm has been discovered
for evaluating an arbitrary AND-OR tree on N variables [4, 13].
Open Problems
In several applications of the quantum search algorithm, only the leading asymptotic behavior
of the query complexity is known, and attempts to suppress logarithmic corrections (when they
appear) and reduce the scaling constants continue [34]. In this section, we point out some other
offbeat applications.
Hamiltonian evolution: Many conventional algorithms for simulations of quantum systems
with sparse Hamiltonians use the Trotter formula with a small step size. They have a power-law
dependence of the computational complexity on the simulation error and hence are not efficient.
In contrast, Grover search amounts to a Trotter formula with the largest possible step size, given
the projection operator nature of the terms in the Hamiltonian. A recent exciting realization is
that this feature leads to only logarithmic dependence of the computational complexity on the
simulation error, which is an exponential improvement. The general strategy is to decompose the
sparse Hamiltonian as a sum of projection operators, formulate the evolution problem as a multiquery search problem, and then use a large step size Trotter formula to simulate it [7, 31]. This
framework can also readily benefit classical simulations of quantum systems.
Molecular biology: Many molecular processes of metabolism occur at scales, nanometer and
picosecond, where quantum dynamics is relevant. They frequently involve unstructured search and
transport, in the sense that correct ingredients for the processes have to be found from the mixture
of molecules floating around. Evolution over billions of years has certainly produced complex
machinery to carry out these searches efficiently, although we do not fully comprehend their
optimization criteria. Attempts to understand some of these processes suggest that Grover search
may have played a role in their design, quite likely exploiting coherent coupled vibrational modes
and not quantum superposition. An intriguing example is that the universal genetic language uses
an alphabet of four letters, while a binary alphabet would be sufficient and simpler to construct
during evolution [30]. Coherent vibrational dynamics of molecules also contributes to efficient
energy transport during photosynthesis and to the detection of smell [23].
Ordered search: A sequentially ordered database can be easily searched by factoring f .i / into
subqueries for individual digits of i . An alternative is to use a different oracle g.i /, such that
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g.i / D 0 when i represents items before the desired item and g.i / D 1 otherwise. Classically,
binary search is the optimal algorithm given either f .i / or g.i / and requires dlog2 N e queries. The
optimal quantum algorithm for f .i / is quaternary search with 0:5dlog2 N e queries, but surprisingly
a quantum algorithm using g.i / can do better. In case of g.i /, though the optimal solution is
unknown, the query complexity for an exact algorithm has a lower bound of 0:221 log2 N [21] and
a known solution of 0:433 log2 N [12] (there also exists a quantum stochastic Las Vegas algorithm
with 0:32 log2 N expected queries and o.1/ error [6]).
Search with additional structure: It may be possible to speed up a search process beyond the
square-root speedup of Grover search, when the problem has extra structure beyond the minimal
information provided by the oracle f .i /. The details of the algorithm and the extent of speedup
would then depend on the extra structure, and the possibilities are open to explorations. Some
examples are symmetries among the items, associative memory recall with connections, and
patterns in the Boolean function to be evaluated. Another problem of interest is determination
of the complete path (with certain properties) from the initial to the target state instead of locations
of just the end points.
Perspective: In a lecture at the Bell Labs in 1985, Richard Feynman made an interesting
observation. In the 1940s when airplanes were being developed, aeronautical engineers had proved
bounds and theorems about why planes would never be able to fly faster than the speed of sound.
For several years, this speed was regarded as fundamentally a bound for flights as the speed of light
is for communications. However, gradually just by using intelligent design, it was discovered that
airplanes could indeed fly faster than the speed of sound – only the rules of design in the new regime
were
pvery
different. The question is whether the bounds on quantum computation (specifically the
N bound for search) will continue to hold, or by making the rules of design very different,
just as in the case of supersonic airplanes, someone will find a way around these bounds. No
one has found any loophole in the arguments in the 20 years since the lower bound for quantum
search was discovered, despite numerous scientists from different fields having tried their hand at
it. On the other hand, even though this bound has been derived over and over again using different
methods, no one has come up with a simple and short physical explanation for it, which would
give one the assurance that one really understood it.
Cross-References
Quantum
Algorithm for Element Distinctness
Routing
Recommended Reading
1. Aaronson S, Ambainis A (2005) Quantum search of spatial regions. Theory Comput 1:47–79
2. Ambainis A (2007) Quantum walk algorithm for element distinctness. SIAM J Comput
37(1):210–239
3. Ambainis A (2010) Quantum search with variable times. Theory Comput Syst 47(3):
786–807
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