140 research outputs found
A logarithmic-depth quantum carry-lookahead adder
We present an efficient addition circuit, borrowing techniques from the
classical carry-lookahead arithmetic circuit. Our quantum carry-lookahead
(QCLA) adder accepts two n-bit numbers and adds them in O(log n) depth using
O(n) ancillary qubits. We present both in-place and out-of-place versions, as
well as versions that add modulo 2^n and modulo 2^n - 1.
Previously, the linear-depth ripple-carry addition circuit has been the
method of choice. Our work reduces the cost of addition dramatically with only
a slight increase in the number of required qubits. The QCLA adder can be used
within current modular multiplication circuits to reduce substantially the
run-time of Shor's algorithm.Comment: 21 pages, 4 color figure
On the Effect of Quantum Interaction Distance on Quantum Addition Circuits
We investigate the theoretical limits of the effect of the quantum
interaction distance on the speed of exact quantum addition circuits. For this
study, we exploit graph embedding for quantum circuit analysis. We study a
logical mapping of qubits and gates of any -depth quantum adder
circuit for two -qubit registers onto a practical architecture, which limits
interaction distance to the nearest neighbors only and supports only one- and
two-qubit logical gates. Unfortunately, on the chosen -dimensional practical
architecture, we prove that the depth lower bound of any exact quantum addition
circuits is no longer , but . This
result, the first application of graph embedding to quantum circuits and
devices, provides a new tool for compiler development, emphasizes the impact of
quantum computer architecture on performance, and acts as a cautionary note
when evaluating the time performance of quantum algorithms.Comment: accepted for ACM Journal on Emerging Technologies in Computing
System
Arithmetic on a Distributed-Memory Quantum Multicomputer
We evaluate the performance of quantum arithmetic algorithms run on a
distributed quantum computer (a quantum multicomputer). We vary the node
capacity and I/O capabilities, and the network topology. The tradeoff of
choosing between gates executed remotely, through ``teleported gates'' on
entangled pairs of qubits (telegate), versus exchanging the relevant qubits via
quantum teleportation, then executing the algorithm using local gates
(teledata), is examined. We show that the teledata approach performs better,
and that carry-ripple adders perform well when the teleportation block is
decomposed so that the key quantum operations can be parallelized. A node size
of only a few logical qubits performs adequately provided that the nodes have
two transceiver qubits. A linear network topology performs acceptably for a
broad range of system sizes and performance parameters. We therefore recommend
pursuing small, high-I/O bandwidth nodes and a simple network. Such a machine
will run Shor's algorithm for factoring large numbers efficiently.Comment: 24 pages, 10 figures, ACM transactions format. Extended version of
Int. Symp. on Comp. Architecture (ISCA) paper; v2, correct one circuit error,
numerous small changes for clarity, add reference
Fast Quantum Modular Exponentiation
We present a detailed analysis of the impact on modular exponentiation of
architectural features and possible concurrent gate execution. Various
arithmetic algorithms are evaluated for execution time, potential concurrency,
and space tradeoffs. We find that, to exponentiate an n-bit number, for storage
space 100n (twenty times the minimum 5n), we can execute modular exponentiation
two hundred to seven hundred times faster than optimized versions of the basic
algorithms, depending on architecture, for n=128. Addition on a neighbor-only
architecture is limited to O(n) time when non-neighbor architectures can reach
O(log n), demonstrating that physical characteristics of a computing device
have an important impact on both real-world running time and asymptotic
behavior. Our results will help guide experimental implementations of quantum
algorithms and devices.Comment: to appear in PRA 71(5); RevTeX, 12 pages, 12 figures; v2 revision is
substantial, with new algorithmic variants, much shorter and clearer text,
and revised equation formattin
On the realistic worst case analysis of quantum arithmetic circuits
We provide evidence that commonly held intuitions when designing quantum
circuits can be misleading. In particular we show that: a) reducing the T-count
can increase the total depth; b) it may be beneficial to trade CNOTs for
measurements in NISQ circuits; c) measurement-based uncomputation of relative
phase Toffoli ancillae can make up to 30\% of a circuit's depth; d) area and
volume cost metrics can misreport the resource analysis. Our findings assume
that qubits are and will remain a very scarce resource. The results are
applicable for both NISQ and QECC protected circuits. Our method uses multiple
ways of decomposing Toffoli gates into Clifford+T gates. We illustrate our
method on addition and multiplication circuits using ripple-carry. As a
byproduct result we show systematically that for a practically significant
range of circuit widths, ripple-carry addition circuits are more resource
efficient than the carry-lookahead addition ones. The methods and circuits were
implemented in the open-source QUANTIFY software
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