729 research outputs found
Efficient Approximation of Diagonal Unitaries over the Clifford+T Basis
We present an algorithm for the approximate decomposition of diagonal
operators, focusing specifically on decompositions over the Clifford+ basis,
that minimize the number of phase-rotation gates in the synthesized
approximation circuit. The equivalent -count of the synthesized circuit is
bounded by , where is the number
of distinct phases in the diagonal -qubit unitary, is the
desired precision, is a quality factor of the implementation method
(), and is the total entanglement cost (in gates). We
determine an optimal decision boundary in -space where our
decomposition algorithm achieves lower entanglement cost than previous
state-of-the-art techniques. Our method outperforms state-of-the-art techniques
for a practical range of values and diagonal operators and can
reduce the number of gates exponentially in when .Comment: 18 pages, 8 figures; introduction improved for readability,
references added (in particular to Dawson & Nielsen
A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits
We present an algorithm for computing depth-optimal decompositions of logical
operations, leveraging a meet-in-the-middle technique to provide a significant
speed-up over simple brute force algorithms. As an illustration of our method
we implemented this algorithm and found factorizations of the commonly used
quantum logical operations into elementary gates in the Clifford+T set. In
particular, we report a decomposition of the Toffoli gate over the set of
Clifford and T gates. Our decomposition achieves a total T-depth of 3, thereby
providing a 40% reduction over the previously best known decomposition for the
Toffoli gate. Due to the size of the search space the algorithm is only
practical for small parameters, such as the number of qubits, and the number of
gates in an optimal implementation.Comment: 23 pages, 15 figures, 1 table; To appear in IEEE Transactions on
Computer-Aided Design of Integrated Circuits and System
Asymptotically optimal approximation of single qubit unitaries by Clifford and T circuits using a constant number of ancillary qubits
We present an algorithm for building a circuit that approximates single qubit
unitaries with precision {\epsilon} using O(log(1/{\epsilon})) Clifford and T
gates and employing up to two ancillary qubits. The algorithm for computing our
approximating circuit requires an average of O(log^2(1/{\epsilon})log
log(1/{\epsilon})) operations. We prove that the number of gates in our circuit
saturates the lower bound on the number of gates required in the scenario when
a constant number of ancillae are supplied, and as such, our circuits are
asymptotically optimal. This results in significant improvement over the
current state of the art for finding an approximation of a unitary, including
the Solovay-Kitaev algorithm that requires O(log^{3+{\delta}}(1/{\epsilon}))
gates and does not use ancillae and the phase kickback approach that requires
O(log^2(1/{\epsilon})log log(1/{\epsilon})) gates, but uses
O(log^2(1/{\epsilon})) ancillae
Efficient synthesis of probabilistic quantum circuits with fallback
Recently it has been shown that Repeat-Until-Success (RUS) circuits can
approximate a given single-qubit unitary with an expected number of gates
of about of what is required by optimal, deterministic, ancilla-free
decompositions over the Clifford+ gate set. In this work, we introduce a
more general and conceptually simpler circuit decomposition method that allows
for synthesis into protocols that probabilistically implement quantum circuits
over several universal gate sets including, but not restricted to, the
Clifford+ gate set. The protocol, which we call Probabilistic Quantum
Circuits with Fallback (PQF), implements a walk on a discrete Markov chain in
which the target unitary is an absorbing state and in which transitions are
induced by multi-qubit unitaries followed by measurements. In contrast to RUS
protocols, the presented PQF protocols terminate after a finite number of
steps. Specifically, we apply our method to the Clifford+, Clifford+, and
Clifford+ gate sets to achieve decompositions with expected gate counts
of , where is a
quantity related to the expansion property of the underlying universal gate
set.Comment: 17 pages, 7 figures; added Appendix F on the runtime performance of
the synthesis algorith
Efficient Decomposition of Single-Qubit Gates into Basis Circuits
We develop the first constructive algorithms for compiling single-qubit
unitary gates into circuits over the universal basis. The basis is an
alternative universal basis to the more commonly studied basis. We
propose two classical algorithms for quantum circuit compilation: the first
algorithm has expected polynomial time (in precision ) and
offers a depth/precision guarantee that improves upon state-of-the-art methods
for compiling into the basis by factors ranging from 1.86 to
. The second algorithm is analogous to direct search and yields
circuits a factor of 3 to 4 times shorter than our first algorithm, and
requires time exponential in ; however, we show that in
practice the runtime is reasonable for an important range of target precisions.Comment: 13 page
Parallelizing quantum circuit synthesis
Quantum circuit synthesis is the process in which an arbitrary unitary
operation is decomposed into a sequence of gates from a universal set,
typically one which a quantum computer can implement both efficiently and
fault-tolerantly. As physical implementations of quantum computers improve, the
need is growing for tools which can effectively synthesize components of the
circuits and algorithms they will run. Existing algorithms for exact,
multi-qubit circuit synthesis scale exponentially in the number of qubits and
circuit depth, leaving synthesis intractable for circuits on more than a
handful of qubits. Even modest improvements in circuit synthesis procedures may
lead to significant advances, pushing forward the boundaries of not only the
size of solvable circuit synthesis problems, but also in what can be realized
physically as a result of having more efficient circuits.
We present a method for quantum circuit synthesis using deterministic walks.
Also termed pseudorandom walks, these are walks in which once a starting point
is chosen, its path is completely determined. We apply our method to construct
a parallel framework for circuit synthesis, and implement one such version
performing optimal -count synthesis over the Clifford+ gate set. We use
our software to present examples where parallelization offers a significant
speedup on the runtime, as well as directly confirm that the 4-qubit 1-bit full
adder has optimal -count 7 and -depth 3.Comment: 16 pages, 9 figure
Floating Point Representations in Quantum Circuit Synthesis
We provide a non-deterministic quantum protocol that approximates the single
qubit rotations R_x(2a^2 b^2)$ using R_x(2a) and R_x(2b) and a constant number
of Clifford and T operations. We then use this method to construct a "floating
point" implementation of a small rotation wherein we use the aforementioned
method to construct the exponent part of the rotation and also to combine it
with a mantissa. This causes the cost of the synthesis to depend more strongly
on the relative (rather than absolute) precision required. We analyze the mean
and variance of the \Tcount required to use our techniques and provide new
lower bounds for the T-count for ancilla free synthesis of small single-qubit
axial rotations. We further show that our techniques can use ancillas to beat
these lower bounds with high probability. We also discuss the T-depth of our
method and see that the vast majority of the cost of the resultant circuits can
be shifted to parallel computation paths.Comment: Comments welcom
A Framework for Approximating Qubit Unitaries
We present an algorithm for efficiently approximating of qubit unitaries over
gate sets derived from totally definite quaternion algebras. It achieves
-approximations using circuits of length ,
which is asymptotically optimal. The algorithm achieves the same quality of
approximation as previously-known algorithms for Clifford+T [arXiv:1212.6253],
V-basis [arXiv:1303.1411] and Clifford+ [arXiv:1409.3552], running on
average in time polynomial in (conditional on a
number-theoretic conjecture). Ours is the first such algorithm that works for a
wide range of gate sets and provides insight into what should constitute a
"good" gate set for a fault-tolerant quantum computer.Comment: 60 pages, 16 figure
Efficient synthesis of universal Repeat-Until-Success circuits
Recently, it was shown that Repeat-Until-Success (RUS) circuits can achieve a
times reduction in expected -count over ancilla-free techniques for
single-qubit unitary decomposition. However, the previously best known
algorithm to synthesize RUS circuits requires exponential classical runtime. In
this paper we present an algorithm to synthesize an RUS circuit to approximate
any given single-qubit unitary within precision in
probabilistically polynomial classical runtime. Our synthesis approach uses the
Clifford+ basis, plus one ancilla qubit and measurement. We provide
numerical evidence that our RUS circuits have an expected -count on average
times lower than the theoretical lower bound of for ancilla-free single-qubit circuit decomposition.Comment: 15 pages, 10 figures; reformatted and minor edits; added Fig. 2 to
visualize the density of z-rotations implementable via RUS protocol
A Depth-Optimal Canonical Form for Single-qubit Quantum Circuits
Given an arbitrary single-qubit operation, an important task is to
efficiently decompose this operation into an (exact or approximate) sequence of
fault-tolerant quantum operations. We derive a depth-optimal canonical form for
single-qubit quantum circuits, and the corresponding rules for exactly reducing
an arbitrary single-qubit circuit to this canonical form. We focus on the
single-qubit universal H,T basis due to its role in fault-tolerant quantum
computing, and show how our formalism might be extended to other universal
bases. We then extend our canonical representation to the family of
Solovay-Kitaev decomposition algorithms, in order to find an
\epsilon-approximation to the single-qubit circuit in polylogarithmic time. For
a given single-qubit operation, we find significantly lower-depth
\epsilon-approximation circuits than previous state-of-the-art implementations.
In addition, the implementation of our algorithm requires significantly fewer
resources, in terms of computation memory, than previous approaches.Comment: 10 pages, 3 figure
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