14 research outputs found
Quantum and Classical Tradeoffs
We propose an approach for quantifying a quantum circuit's quantumness as a
means to understand the nature of quantum algorithmic speedups. Since quantum
gates that do not preserve the computational basis are necessary for achieving
quantum speedups, it appears natural to define the quantumness of a quantum
circuit using the number of such gates. Intuitively, a reduction in the
quantumness requires an increase in the amount of classical computation, hence
giving a ``quantum and classical tradeoff''.
In this paper we present two results on this direction. The first gives an
asymptotic answer to the question: ``what is the minimum number of
non-basis-preserving gates required to generate a good approximation to a given
state''. This question is the quantum analogy of the following classical
question, ``how many fair coins are needed to generate a given probability
distribution'', which was studied and resolved by Knuth and Yao in 1976. Our
second result shows that any quantum algorithm that solves Grover's Problem of
size n using k queries and l levels of non-basis-preserving gates must have
k*l=\Omega(n)
On the Role of Hadamard Gates in Quantum Circuits
We study a reduced quantum circuit computation paradigm in which the only
allowable gates either permute the computational basis states or else apply a
"global Hadamard operation", i.e. apply a Hadamard operation to every qubit
simultaneously. In this model, we discuss complexity bounds (lower-bounding the
number of global Hadamard operations) for common quantum algorithms : we
illustrate upper bounds for Shor's Algorithm, and prove lower bounds for
Grover's Algorithm. We also use our formalism to display a gate that is neither
quantum-universal nor classically simulable, on the assumption that Integer
Factoring is not in BPP.Comment: 16 pages, last section clarified, typos corrected, references added,
minor rewordin
NP-hard problems are not in BQP
Grover's algorithm can solve NP-complete problems on quantum computers faster
than all the known algorithms on classical computers. However, Grover's
algorithm still needs exponential time. Due to the BBBV theorem, Grover's
algorithm is optimal for searches in the domain of a function, when the
function is used as a black box.
We analyze the NP-complete set
If is large enough, then M accepts each word in with length
within steps. So, one can use methods from computability theory to show
that black box searching is the fastest way to find a solution. Therefore,
Grover's algorithm is optimal for NP-complete problems
Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy
We introduce a simple sub-universal quantum computing model, which we call
the Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of a
classical reversible circuit sandwiched by two layers of Hadamard gates, and
therefore it is in the second level of the Fourier hierarchy. We show that
output probability distributions of the HC1Q model cannot be classically
efficiently sampled within a multiplicative error unless the polynomial-time
hierarchy collapses to the second level. The proof technique is different from
those used for previous sub-universal models, such as IQP, Boson Sampling, and
DQC1, and therefore the technique itself might be useful for finding other
sub-universal models that are hard to classically simulate. We also study the
classical verification of quantum computing in the second level of the Fourier
hierarchy. To this end, we define a promise problem, which we call the
probability distribution distinguishability with maximum norm (PDD-Max). It is
a promise problem to decide whether output probability distributions of two
quantum circuits are far apart or close. We show that PDD-Max is BQP-complete,
but if the two circuits are restricted to some types in the second level of the
Fourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has a
Merlin-Arthur system with quantum polynomial-time Merlin and classical
probabilistic polynomial-time Arthur.Comment: 30 pages, 4 figure
The Quantum Frontier
The success of the abstract model of computation, in terms of bits, logical
operations, programming language constructs, and the like, makes it easy to
forget that computation is a physical process. Our cherished notions of
computation and information are grounded in classical mechanics, but the
physics underlying our world is quantum. In the early 80s researchers began to
ask how computation would change if we adopted a quantum mechanical, instead of
a classical mechanical, view of computation. Slowly, a new picture of
computation arose, one that gave rise to a variety of faster algorithms, novel
cryptographic mechanisms, and alternative methods of communication. Small
quantum information processing devices have been built, and efforts are
underway to build larger ones. Even apart from the existence of these devices,
the quantum view on information processing has provided significant insight
into the nature of computation and information, and a deeper understanding of
the physics of our universe and its connections with computation.
We start by describing aspects of quantum mechanics that are at the heart of
a quantum view of information processing. We give our own idiosyncratic view of
a number of these topics in the hopes of correcting common misconceptions and
highlighting aspects that are often overlooked. A number of the phenomena
described were initially viewed as oddities of quantum mechanics. It was
quantum information processing, first quantum cryptography and then, more
dramatically, quantum computing, that turned the tables and showed that these
oddities could be put to practical effect. It is these application we describe
next. We conclude with a section describing some of the many questions left for
future work, especially the mysteries surrounding where the power of quantum
information ultimately comes from.Comment: Invited book chapter for Computation for Humanity - Information
Technology to Advance Society to be published by CRC Press. Concepts
clarified and style made more uniform in version 2. Many thanks to the
referees for their suggestions for improvement
Delegating Quantum Computation in the Quantum Random Oracle Model
A delegation scheme allows a computationally weak client to use a server's
resources to help it evaluate a complex circuit without leaking any information
about the input (other than its length) to the server. In this paper, we
consider delegation schemes for quantum circuits, where we try to minimize the
quantum operations needed by the client. We construct a new scheme for
delegating a large circuit family, which we call "C+P circuits". "C+P" circuits
are the circuits composed of Toffoli gates and diagonal gates. Our scheme is
non-interactive, requires very little quantum computation from the client
(proportional to input length but independent of the circuit size), and can be
proved secure in the quantum random oracle model, without relying on additional
assumptions, such as the existence of fully homomorphic encryption. In practice
the random oracle can be replaced by an appropriate hash function or block
cipher, for example, SHA-3, AES.
This protocol allows a client to delegate the most expensive part of some
quantum algorithms, for example, Shor's algorithm. The previous protocols that
are powerful enough to delegate Shor's algorithm require either many rounds of
interactions or the existence of FHE. The protocol requires asymptotically
fewer quantum gates on the client side compared to running Shor's algorithm
locally.
To hide the inputs, our scheme uses an encoding that maps one input qubit to
multiple qubits. We then provide a novel generalization of classical garbled
circuits ("reversible garbled circuits") to allow the computation of Toffoli
circuits on this encoding. We also give a technique that can support the
computation of phase gates on this encoding.
To prove the security of this protocol, we study key dependent message(KDM)
security in the quantum random oracle model. KDM security was not previously
studied in quantum settings.Comment: 41 pages, 1 figures. Update to be consistent with the proceeding
versio
Interactive Proofs with Polynomial-Time Quantum Prover for Computing the Order of Solvable Groups
In this paper we consider what can be computed by a user interacting with a potentially malicious server, when the server performs polynomial-time quantum computation but the user can only perform polynomial-time classical (i.e., non-quantum) computation. Understanding the computational power of this model, which corresponds to polynomial-time quantum computation that can be efficiently verified classically, is a well-known open problem in quantum computing. Our result shows that computing the order of a solvable group, which is one of the most general problems for which quantum computing exhibits an exponential speed-up with respect to classical computing, can be realized in this model