4,768 research outputs found
Black-box Hamiltonian simulation and unitary implementation
We present general methods for simulating black-box Hamiltonians using
quantum walks. These techniques have two main applications: simulating sparse
Hamiltonians and implementing black-box unitary operations. In particular, we
give the best known simulation of sparse Hamiltonians with constant precision.
Our method has complexity linear in both the sparseness D (the maximum number
of nonzero elements in a column) and the evolution time t, whereas previous
methods had complexity scaling as D^4 and were superlinear in t. We also
consider the task of implementing an arbitrary unitary operation given a
black-box description of its matrix elements. Whereas standard methods for
performing an explicitly specified N x N unitary operation use O(N^2)
elementary gates, we show that a black-box unitary can be performed with
bounded error using O(N^{2/3} (log log N)^{4/3}) queries to its matrix
elements. In fact, except for pathological cases, it appears that most
unitaries can be performed with only O(sqrt{N}) queries, which is optimal.Comment: 19 pages, 3 figures, minor correction
Approximating Fractional Time Quantum Evolution
An algorithm is presented for approximating arbitrary powers of a black box
unitary operation, , where is a real number, and
is a black box implementing an unknown unitary. The complexity of
this algorithm is calculated in terms of the number of calls to the black box,
the errors in the approximation, and a certain `gap' parameter. For general
and large , one should apply a total of times followed by our procedure for approximating the fractional
power . An example is also given where for
large integers this method is more efficient than direct application of
copies of . Further applications and related algorithms are also
discussed.Comment: 13 pages, 2 figure
On the relationship between continuous- and discrete-time quantum walk
Quantum walk is one of the main tools for quantum algorithms. Defined by
analogy to classical random walk, a quantum walk is a time-homogeneous quantum
process on a graph. Both random and quantum walks can be defined either in
continuous or discrete time. But whereas a continuous-time random walk can be
obtained as the limit of a sequence of discrete-time random walks, the two
types of quantum walk appear fundamentally different, owing to the need for
extra degrees of freedom in the discrete-time case.
In this article, I describe a precise correspondence between continuous- and
discrete-time quantum walks on arbitrary graphs. Using this correspondence, I
show that continuous-time quantum walk can be obtained as an appropriate limit
of discrete-time quantum walks. The correspondence also leads to a new
technique for simulating Hamiltonian dynamics, giving efficient simulations
even in cases where the Hamiltonian is not sparse. The complexity of the
simulation is linear in the total evolution time, an improvement over
simulations based on high-order approximations of the Lie product formula. As
applications, I describe a continuous-time quantum walk algorithm for element
distinctness and show how to optimally simulate continuous-time query
algorithms of a certain form in the conventional quantum query model. Finally,
I discuss limitations of the method for simulating Hamiltonians with negative
matrix elements, and present two problems that motivate attempting to
circumvent these limitations.Comment: 22 pages. v2: improved presentation, new section on Hamiltonian
oracles; v3: published version, with improved analysis of phase estimatio
Exponential improvement in precision for simulating sparse Hamiltonians
We provide a quantum algorithm for simulating the dynamics of sparse
Hamiltonians with complexity sublogarithmic in the inverse error, an
exponential improvement over previous methods. Specifically, we show that a
-sparse Hamiltonian acting on qubits can be simulated for time
with precision using queries and
additional 2-qubit gates, where . Unlike previous
approaches based on product formulas, the query complexity is independent of
the number of qubits acted on, and for time-varying Hamiltonians, the gate
complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our
algorithm is based on a significantly improved simulation of the continuous-
and fractional-query models using discrete quantum queries, showing that the
former models are not much more powerful than the discrete model even for very
small error. We also simplify the analysis of this conversion, avoiding the
need for a complex fault correction procedure. Our simplification relies on a
new form of "oblivious amplitude amplification" that can be applied even though
the reflection about the input state is unavailable. Finally, we prove new
lower bounds showing that our algorithms are optimal as a function of the
error.Comment: v1: 27 pages; Subsumes and improves upon results in arXiv:1308.5424.
v2: 28 pages, minor change
Simulating sparse Hamiltonians with star decompositions
We present an efficient algorithm for simulating the time evolution due to a
sparse Hamiltonian. In terms of the maximum degree d and dimension N of the
space on which the Hamiltonian H acts for time t, this algorithm uses
(d^2(d+log* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the
sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders,
which scales like (d^4(log* N)||Ht||)^{1+o(1)}. To achieve this, we decompose a
general sparse Hamiltonian into a small sum of Hamiltonians whose graphs of
non-zero entries have the property that every connected component is a star,
and efficiently simulate each of these pieces.Comment: 11 pages. v2: minor correction
Efficient discrete-time simulations of continuous-time quantum query algorithms
The continuous-time query model is a variant of the discrete query model in
which queries can be interleaved with known operations (called "driving
operations") continuously in time. Interesting algorithms have been discovered
in this model, such as an algorithm for evaluating nand trees more efficiently
than any classical algorithm. Subsequent work has shown that there also exists
an efficient algorithm for nand trees in the discrete query model; however,
there is no efficient conversion known for continuous-time query algorithms for
arbitrary problems.
We show that any quantum algorithm in the continuous-time query model whose
total query time is T can be simulated by a quantum algorithm in the discrete
query model that makes O[T log(T) / log(log(T))] queries. This is the first
upper bound that is independent of the driving operations (i.e., it holds even
if the norm of the driving Hamiltonian is very large). A corollary is that any
lower bound of T queries for a problem in the discrete-time query model
immediately carries over to a lower bound of \Omega[T log(log(T))/log (T)] in
the continuous-time query model.Comment: 12 pages, 6 fig
Exponential algorithmic speedup by quantum walk
We construct an oracular (i.e., black box) problem that can be solved
exponentially faster on a quantum computer than on a classical computer. The
quantum algorithm is based on a continuous time quantum walk, and thus employs
a different technique from previous quantum algorithms based on quantum Fourier
transforms. We show how to implement the quantum walk efficiently in our
oracular setting. We then show how this quantum walk can be used to solve our
problem by rapidly traversing a graph. Finally, we prove that no classical
algorithm can solve this problem with high probability in subexponential time.Comment: 24 pages, 7 figures; minor corrections and clarification
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