15,555 research outputs found
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
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
Quantum arithmetic and numerical analysis using Repeat-Until-Success circuits
We develop a method for approximate synthesis of single--qubit rotations of
the form that is based on the
Repeat-Until-Success (RUS) framework for quantum circuit synthesis. We
demonstrate how smooth computable functions can be synthesized from two
basic primitives. This synthesis approach constitutes a manifestly quantum form
of arithmetic that differs greatly from the approaches commonly used in quantum
algorithms. The key advantage of our approach is that it requires far fewer
qubits than existing approaches: as a case in point, we show that using as few
as ancilla qubits, one can obtain RUS circuits for approximate
multiplication and reciprocals. We also analyze the costs of performing
multiplication and inversion on a quantum computer using conventional
approaches and find that they can require too many qubits to execute on a small
quantum computer, unlike our approach
Factoring with Qutrits: Shor's Algorithm on Ternary and Metaplectic Quantum Architectures
We determine the cost of performing Shor's algorithm for integer
factorization on a ternary quantum computer, using two natural models of
universal fault-tolerant computing:
(i) a model based on magic state distillation that assumes the availability
of the ternary Clifford gates, projective measurements, classical control as
its natural instrumentation set; (ii) a model based on a metaplectic
topological quantum computer (MTQC). A natural choice to implement Shor's
algorithm on a ternary quantum computer is to translate the entire arithmetic
into a ternary form. However, it is also possible to emulate the standard
binary version of the algorithm by encoding each qubit in a three-level system.
We compare the two approaches and analyze the complexity of implementing Shor's
period finding function in the two models. We also highlight the fact that the
cost of achieving universality through magic states in MTQC architecture is
asymptotically lower than in generic ternary case.Comment: 22 pages, 7 figures; v3: significant overhaul; this version focuses
on the use of true ternary vs. emulated binary encodin
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
Repeat-Until-Success circuits with fixed-point oblivious amplitude amplification
Certain quantum operations can be built more efficiently through a procedure
known as Repeat-Until-Success. Differently from other non-deterministic quantum
operations, this procedure provides a classical flag which certifies the
success or failure of the procedure and, in the latter case, a recovery step
allows the restoration of the quantum state to its original condition. The
procedure can then be repeated until success is achieved. After success is
certified, the RUS procedure can be equated to a coherent gate. However, this
is not the case when the operation needs to be conditioned on the state of
other qubits, possibly being in a superposition state. In this situation, the
final operation depends on the failure and success history and introduces a
"distortion" that, even after the final success, depends on the past outcomes.
We quantify the distortion and show that it can be reduced by increasing the
probability of success towards unity. While this can be achieved via oblivious
amplitude amplification when the initial success probability is known, we
propose the use of fixed-point oblivious amplitude amplification to reduce this
unwanted distortions below any given threshold even without knowing the initial
success probability
Scalable randomized benchmarking of non-Clifford gates
Randomized benchmarking is a widely used experimental technique to
characterize the average error of quantum operations. Benchmarking procedures
that scale to enable characterization of -qubit circuits rely on efficient
procedures for manipulating those circuits and, as such, have been limited to
subgroups of the Clifford group. However, universal quantum computers require
additional, non-Clifford gates to approximate arbitrary unitary
transformations. We define a scalable randomized benchmarking procedure over
-qubit unitary matrices that correspond to protected non-Clifford gates for
a class of stabilizer codes. We present efficient methods for representing and
composing group elements, sampling them uniformly, and synthesizing
corresponding -sized circuits. The procedure provides
experimental access to two independent parameters that together characterize
the average gate fidelity of a group element.Comment: 5+4 pages, 1 figur
T-count optimization and Reed-Muller codes
In this paper, we study the close relationship between Reed-Muller codes and
single-qubit phase gates from the perspective of -count optimization. We
prove that minimizing the number of gates in an -qubit quantum circuit
over CNOT and , together with the Clifford group powers of , corresponds
to finding a minimum distance decoding of a length binary vector in the
order punctured Reed-Muller code. Moreover, we show that the problems are
polynomially equivalent in the length of the code. As a consequence, we derive
an algorithm for the optimization of -count in quantum circuits based on
Reed-Muller decoders, along with a new upper bound of on the number of
gates required to implement an -qubit unitary over CNOT and gates.
We further generalize this result to show that minimizing small angle rotations
corresponds to decoding lower order binary Reed-Muller codes. In particular, we
show that minimizing the number of gates for any integer is
equivalent to minimum distance decoding in ,
where is the highest power of dividing .Comment: 19 pages. Version 2 gives a substantially different presentation of
the results, as well as a generalization to rotation angles of any finite
orde
Advantages of a modular high-level quantum programming framework
We review some of the features of the ProjectQ software framework and
quantify their impact on the resulting circuits. The concise high-level
language facilitates implementing even complex algorithms in a very
time-efficient manner while, at the same time, providing the compiler with
additional information for optimization through code annotation - so-called
meta-instructions. We investigate the impact of these annotations for the
example of Shor's algorithm in terms of logical gate counts. Furthermore, we
analyze the effect of different intermediate gate sets for optimization and how
the dimensions of the resulting circuit depend on a smart choice thereof.
Finally, we demonstrate the benefits of a modular compilation framework by
implementing mapping procedures for one- and two-dimensional nearest neighbor
architectures which we then compare in terms of overhead for different problem
sizes
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
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