67 research outputs found
A 2D Nearest-Neighbor Quantum Architecture for Factoring in Polylogarithmic Depth
We contribute a 2D nearest-neighbor quantum architecture for Shor's algorithm
to factor an -bit number in depth. Our implementation uses
parallel phase estimation, constant-depth fanout and teleportation, and
constant-depth carry-save modular addition. We derive upper bounds on the
circuit resources of our architecture under a new 2D nearest-neighbor model
which allows a classical controller and parallel, communicating modules. We
also contribute a novel constant-depth circuit for unbounded quantum unfanout
in our new model. Finally, we provide a comparison to all previous
nearest-neighbor factoring implementations. Our circuit results in an
exponential improvement in nearest-neighbor circuit depth at the cost of a
polynomial increase in circuit size and width.Comment: 29 pages, 14 figures, 3 tables, presented at Reversible Computation
Workshop 2012 in Copenhagen. Updated with numerical circuit resource upper
bounds and constant-depth quantum unfanou
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
Low-distance Surface Codes under Realistic Quantum Noise
We study the performance of distance-three surface code layouts under
realistic multi-parameter noise models. We first calculate their thresholds
under depolarizing noise. We then compare a Pauli-twirl approximation of
amplitude and phase damping to amplitude and phase damping. We find the
approximate channel results in a pessimistic estimate of the logical error
rate, indicating the realistic threshold may be higher than previously
estimated. From Monte-Carlo simulations, we identify experimental parameters
for which these layouts admit reliable computation. Due to its low resource
cost and superior performance, we conclude that the 17-qubit layout should be
targeted in early experimental implementations of the surface code. We find
that architectures with gate times in the 5-40 ns range and T1 times of at
least 1-2 us range will exhibit improved logical error rates with a 17-qubit
surface code encoding.Comment: 15 pages, 15 figures, 4 tables, comments welcom
Compiling Quantum Circuits using the Palindrome Transform
The design and optimization of quantum circuits is central to quantum
computation. This paper presents new algorithms for compiling arbitrary 2^n x
2^n unitary matrices into efficient circuits of (n-1)-controlled single-qubit
and (n-1)-controlled-NOT gates. We first present a general algebraic
optimization technique, which we call the Palindrome Transform, that can be
used to minimize the number of self-inverting gates in quantum circuits
consisting of concatenations of palindromic subcircuits. For a fixed column
ordering of two-level decomposition, we then give an numerative algorithm for
minimal (n-1)-controlled-NOT circuit construction, which we call the
Palindromic Optimization Algorithm. Our work dramatically reduces the number of
gates generated by the conventional two-level decomposition method for
constructing quantum circuits of (n-1)-controlled single-qubit and
(n-1)-controlled-NOT gates.Comment: 17 pages, LaTe
A State Distillation Protocol to Implement Arbitrary Single-qubit Rotations
An important task required to build a scalable, fault-tolerant quantum
computer is to efficiently represent an arbitrary single-qubit rotation by
fault-tolerant quantum operations. Traditionally, the method for decomposing a
single-qubit unitary into a discrete set of gates is Solovay-Kitaev
decomposition, which in practice produces a sequence of depth
O(\log^c(1/\epsilon)), where c~3.97 is the state-of-the-art. The proven lower
bound is c=1, however an efficient algorithm that saturates this bound is
unknown. In this paper, we present an alternative to Solovay-Kitaev
decomposition employing state distillation techniques which reduces c to
between 1.12 and 2.27, depending on the setting. For a given single-qubit
rotation, our protocol significantly lowers the length of the approximating
sequence and the number of required resource states (ancillary qubits). In
addition, our protocol is robust to noise in the resource states.Comment: 10 pages, 18 figures, 5 table
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
Asymptotically Optimal Topological Quantum Compiling
In a topological quantum computer, universality is achieved by braiding and
quantum information is natively protected from small local errors. We address
the problem of compiling single-qubit quantum operations into braid
representations for non-abelian quasiparticles described by the Fibonacci anyon
model. We develop a probabilistically polynomial algorithm that outputs a braid
pattern to approximate a given single-qubit unitary to a desired precision. We
also classify the single-qubit unitaries that can be implemented exactly by a
Fibonacci anyon braid pattern and present an efficient algorithm to produce
their braid patterns. Our techniques produce braid patterns that meet the
uniform asymptotic lower bound on the compiled circuit depth and thus are
depth-optimal asymptotically. Our compiled circuits are significantly shorter
than those output by prior state-of-the-art methods, resulting in improvements
in depth by factors ranging from 20 to 1000 for precisions ranging between
and .Comment: 24 page
Reversible circuit compilation with space constraints
We develop a framework for resource efficient compilation of higher-level
programs into lower-level reversible circuits. Our main focus is on optimizing
the memory footprint of the resulting reversible networks. This is motivated by
the limited availability of qubits for the foreseeable future. We apply three
main techniques to keep the number of required qubits small when computing
classical, irreversible computations by means of reversible networks: first,
wherever possible we allow the compiler to make use of in-place functions to
modify some of the variables. Second, an intermediate representation is
introduced that allows to trace data dependencies within the program, allowing
to clean up qubits early. This realizes an analog to "garbage collection" for
reversible circuits. Third, we use the concept of so-called pebble games to
transform irreversible programs into reversible programs under space
constraints, allowing for data to be erased and recomputed if needed.
We introduce REVS, a compiler for reversible circuits that can translate a
subset of the functional programming language F# into Toffoli networks which
can then be further interpreted for instance in LIQui|>, a domain-specific
language for quantum computing and which is also embedded into F#. We discuss a
number of test cases that illustrate the advantages of our approach including
reversible implementations of SHA-2 and other cryptographic hash-functions,
reversible integer arithmetic, as well as a test-bench of combinational
circuits used in classical circuit synthesis. Compared to Bennett's method,
REVS can reduce space complexity by a factor of or more, while having an
only moderate increase in circuit size as well as in the time it takes to
compile the reversible networks.Comment: 32 pages, 15 figures, 4 table
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
Quantum Perceptron Models
We demonstrate how quantum computation can provide non-trivial improvements
in the computational and statistical complexity of the perceptron model. We
develop two quantum algorithms for perceptron learning. The first algorithm
exploits quantum information processing to determine a separating hyperplane
using a number of steps sublinear in the number of data points , namely
. The second algorithm illustrates how the classical mistake bound
of can be further improved to
through quantum means, where denotes the
margin. Such improvements are achieved through the application of quantum
amplitude amplification to the version space interpretation of the perceptron
model
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