1,839 research outputs found
Computational depth complexity of measurement-based quantum computation
We prove that one-way quantum computations have the same computational power
as quantum circuits with unbounded fan-out. It demonstrates that the one-way
model is not only one of the most promising models of physical realisation, but
also a very powerful model of quantum computation. It confirms and completes
previous results which have pointed out, for some specific problems, a depth
separation between the one-way model and the quantum circuit model. Since
one-way model has the same computational power as unbounded quantum fan-out
circuits, the quantum Fourier transform can be approximated in constant depth
in the one-way model, and thus the factorisation can be done by a polytime
probabilistic classical algorithm which has access to a constant-depth one-way
quantum computer. The extra power of the one-way model, comparing with the
quantum circuit model, comes from its classical-quantum hybrid nature. We show
that this extra power is reduced to the capability to perform unbounded
classical parity gates in constant depth.Comment: 12 page
Constant-depth circuits for Uniformly Controlled Gates and Boolean functions with application to quantum memory circuits
We explore the power of the unbounded Fan-Out gate and the Global Tunable
gates generated by Ising-type Hamiltonians in constructing constant-depth
quantum circuits, with particular attention to quantum memory devices. We
propose two types of constant-depth constructions for implementing Uniformly
Controlled Gates. These gates include the Fan-In gates defined by
for
and , where is a Boolean function. The first of our
constructions is based on computing the one-hot encoding of the control
register , while the second is based on Boolean analysis and
exploits different representations of such as its Fourier expansion. Via
these constructions, we obtain constant-depth circuits for the quantum
counterparts of read-only and read-write memory devices -- Quantum Random
Access Memory (QRAM) and Quantum Random Access Gate (QRAG) -- of memory size
. The implementation based on one-hot encoding requires either
ancillae and Fan-Out gates or
ancillae and Global Tunable gates. On the other hand, the
implementation based on Boolean analysis requires only Global Tunable gates
at the expense of ancillae.Comment: 50 pages, 10 figures. Comments are welcom
Bounds on the Power of Constant-Depth Quantum Circuits
We show that if a language is recognized within certain error bounds by
constant-depth quantum circuits over a finite family of gates, then it is
computable in (classical) polynomial time. In particular, our results imply
EQNC^0 is contained in P, where EQNC^0 is the constant-depth analog of the
class EQP. On the other hand, we adapt and extend ideas of Terhal and
DiVincenzo (quant-ph/0205133) to show that, for any family F of quantum gates
including Hadamard and CNOT gates, computing the acceptance probabilities of
depth-five circuits over F is just as hard as computing these probabilities for
circuits over F. In particular, this implies that NQNC^0 = NQACC = NQP = coC=P
where NQNC^0 is the constant-depth analog of the class NQP. This essentially
refutes a conjecture of Green et al. that NQACC is contained in TC^0
(quant-ph/0106017)
Commuting Quantum Circuits with Few Outputs are Unlikely to be Classically Simulatable
We study the classical simulatability of commuting quantum circuits with n
input qubits and O(log n) output qubits, where a quantum circuit is classically
simulatable if its output probability distribution can be sampled up to an
exponentially small additive error in classical polynomial time. First, we show
that there exists a commuting quantum circuit that is not classically
simulatable unless the polynomial hierarchy collapses to the third level. This
is the first formal evidence that a commuting quantum circuit is not
classically simulatable even when the number of output qubits is exponentially
small. Then, we consider a generalized version of the circuit and clarify the
condition under which it is classically simulatable. Lastly, we apply the
argument for the above evidence to Clifford circuits in a similar setting and
provide evidence that such a circuit augmented by a depth-1 non-Clifford layer
is not classically simulatable. These results reveal subtle differences between
quantum and classical computation.Comment: 19 pages, 6 figures; v2: Theorems 1 and 3 improved, proofs modifie
Small Depth Quantum Circuits
Small depth quantum circuits have proved to be unexpectedly powerful in comparison to their classical counterparts. We survey some of the recent work on this and present some open problems.National Security Agency; Advanced Research and Development Agency under Army Research Office (DAAD 19-02-1-0058
On the power of interleaved low-depth quantum and classical circuits
Low-depth quantum circuits are a well-suited model for near-term quantum devices, given short coherence times and noisy gate operations, making it pivotal to examine their computational power. It was already known as early as 2004 that simulating such low-depth quantum circuits is classically hard under complexity-theoretic assumptions. Later, it was shown that low-depth quantum circuits interleaved with low-depth classical circuits can perform approximate quantum Fourier transform, the quantum subroutine of Shor's algorithm.
Given these salient features of low-depth quantum models, Terhal and DiVincenzo, Aaronson, and Jozsa have all independently conjectured regarding the elusive power of combining low-depth quantum circuits with efficient classical computation. However, much has remained unresolved in this interleaved setting. Therefore, in this thesis, we tackle the question of characterizing the computational power of interleaved low-depth quantum and classical circuits. We first review existing separations in the low-depth setting. Then, we formally define two interleaving models based on whether the quantum device is permitted to make subset measurements (weak interleaving) or must measure all qubits together (strict interleaving).
By combining existing techniques from quantum fan-out constructions, teleportation-based quantum computation, and Clifford + T circuit synthesis, we show several results regarding the power of variants of constant-depth quantum circuits (QNC0) strictly and weakly interleaved with constant-depth classical parity circuits. Our main new result is that QNC0 with access to cat states strictly interleaved with constant-depth classical parity circuits can simulate constant-depth threshold circuits (TC0), which neither of the classes can do on their own. This strictly separates this interleaved class from constant-depth classical circuits with unbounded fan-in mod p and OR gates (AC0[p])
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