Every NAND formula of size N can be evaluated in time N^{1/2+o(1)} on a quantum computer
Andrew M. Childs, Ben W. Reichardt, Robert Spalek, Shengyu Zhang
备注：
14 pages, 3 figures, v2: substantially rewritten with clearer analysis, v3: using better discretization we have obtained an optimal algorithm. to appear in FOCS'07
For every NAND formula of size N, there is a bounded-error N^{1/2+o(1)}-time quantum algorithm, based on a coined quantum walk, that evaluates this formula on a black-box input. Balanced, or ``approximately balanced,'' NAND formulas can be evaluated in O(sqrt{N}) queries, which is optimal. It follows that the (2-o(1))-th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.
Quantum annealing machines based on superconducting qubits, which have the potential to solve optimization problems faster than digital computers, are of great interest not only to researchers but also to the general public. Here, we propose a quantum annealing machine based on a semiconductor floating gate (FG) array. We use the same device structure as that of the commercial FG NAND flash memory except for small differences such as thinner tunneling barrier. We theoretically derive an Ising Hamiltonian from the FG system in its single-electron region. Recent high-density NAND flash memories …
Quantum computers require quantum processors. An important part of the processor of any computer is the arithmetic unit, which performs binary addition, subtraction, division and multiplication, however multiplication can be performed using repeated addition, while division can be performed using repeated subtraction. In this paper we present two designs using the reversible $R^3$ gate to perform the quantum half adder/ subtractor and the quantum full adder/subtractor. The proposed half adder/subtractor design can be used to perform different logical operations, such as $AND$, $XOR$, $NAND$, $…
We give a quantum algorithm for evaluating formulas over an extended gate set, including all two- and three-bit binary gates (e.g., NAND, 3-majority). The algorithm is optimal on read-once formulas for which each gate's inputs are balanced in a certain sense. The main new tool is a correspondence between a classical linear-algebraic model of computation, "span programs," and weighted bipartite graphs. A span program's evaluation corresponds to an eigenvalue-zero eigenvector of the associated graph. A quantum computer can therefore evaluate the span program by applying spectral estimation to th…
Local search algorithms use the neighborhood relations among search states and often perform well for a variety of NP-hard combinatorial search problems. This paper shows how quantum computers can also use these neighborhood relations. An example of such a local quantum search is evaluated empirically for the satisfiability (SAT) problem and shown to be particularly effective for highly constrained instances. For problems with an intermediate number of constraints, it is somewhat less effective at exploiting problem structure than incremental quantum methods, in spite of the much smaller searc…
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The concept of negative probabilities can be used to decompose the interaction of two qubits mediated by a quantum controlled-NOT into three operations that require only classical interactions (that is, local operations and classical communication) between the qubits. For a single gate, the probabilities of the three operations are 1, 1, and -1. This decomposition can be applied in a probabilistic simulation of quantum computation by randomly choosing one of the three operations for each gate and assigning a negative statistical weight to the outcomes of sequences with an odd number of negativ…
Given a blackbox for f, a smooth real scalar function of d real variables, one wants to estimate the gradient of f at a given point with n bits of precision. On a classical computer this requires a minimum of d+1 blackbox queries, whereas on a quantum computer it requires only one query regardless of d. The number of bits of precision to which f must be evaluated matches the classical requirement in the limit of large n.
Calculation aspects of holonomic quantum computer (HQC) are considered. Wilczek--Zee potential defining the set of quantum calculations for HQC is explicitly evaluated. Principal possibility of realization of the logical gates for this case is discussed.
We have studied a system composed by two endohedral fullerene molecules. We have found that this system can be used as good candidate for the realization of Quantum Gates Each of these molecules encapsules an atom carrying a spin,therefore they interact through the spin dipole interaction. We show that a phase gate can be realized if we apply on each encased spin static and time dependent magnetic field. We have evaluated the operational time of a $π$-phase gate, which is of the order of ns. We made a comparison between the theoretical estimation of the gate time and the experimental decoheren…
Certain physical aspects of quantum error correction are discussed for a quantum computer (n-qubit register) in contact with a decohering environment. Under rather plausible assumptions upon the form of the computer-environment interaction, the efficiency of a general correcting procedure is evaluated as a function of the spontaneous-decay duration and the rank of errors covered by the procedure. It is proved that the probability of errors can be made arbitrarily small by enhancing the correction method, provided the decohering interaction is represented by a bounded operator.
The influence of the electric field created by a gate potential of the silicon quantum computer on the hyperfine interaction constant (HIC) is obtained. The errors due to technological inaccuracy of location of donor atoms under a gate are evaluated. The energy spectra of electron-nuclear spin system of two interacting donor atoms with various values of HIC are calculated. The presence of two pairs of anticrossing levels in the ground electronic state is shown. Parameters of the structure at which errors rate can be greatly minimized are found.
The optimal design of a fault-tolerant quantum computer involves finding an appropriate balance between the burden of large-scale integration of noisy components and the load of improving the reliability of hardware technology. This balance can be evaluated by quantitatively modeling the execution of quantum logic operations on a realistic quantum hardware containing limited computational resources. In this work, we report a complete performance simulation software tool capable of (1) searching the hardware design space by varying resource architecture and technology parameters, (2) synthesizi…
CALA-$n$: A Quantum Library for Realizing Cost-Effective 2-, 3-, 4-, and 5-bit Gates on IBM Quantum Computers using Bloch Sphere Approach, Clifford+T Gates, and Layouts
We address the problem of facial expression recognition and show a possible solution using a quantum machine learning approach. In order to define an efficient classifier for a given dataset, our approach substantially exploits quantum interference. By representing face expressions via graphs, we define a classifier as a quantum circuit that manipulates the graphs adjacency matrices encoded into the amplitudes of some appropriately defined quantum states. We discuss the accuracy of the quantum classifier evaluated on the quantum simulator available on the IBM Quantum Experience cloud platform,…
Technion University（以色列理工学院）；The Helen Diller Quantum Center（海伦·迪勒量子中心）；MIGAL – Galilee Research Institute（米加尔-加利利研究所）；Tel-Hai University of Kiryat Shmona and the Galilee（基里亚特谢莫纳和加利利泰尔海大学）
We develop computationally affordable and encoding independent gradient evaluation procedures for unitary coupled-cluster type operators, applicable on quantum computers. We show that, within our framework, the gradient of an expectation value with respect to a parameterized n-fold fermionic excitation can be evaluated by four expectation values of similar form and size, whereas most standard approaches based on the direct application of the parameter-shift-rule come with an associated cost of O(2^(2n)) expectation values. For real wavefunctions, this cost can be further reduced to two expecta…
We propose the first correct special-purpose quantum circuits for preparation of Bell-diagonal states (BDS), and implement them on the IBM Quantum computer, characterizing and testing complex aspects of their quantum correlations in the full parameter space. Among the circuits proposed, one involves only two quantum bits but requires adapted quantum tomography routines handling classical bits in parallel. The entire class of Bell-diagonal states is generated, and a number of characteristic indicators, namely entanglement of formation, CHSH non-locality, steering and discord, are experimentally…
A cloud quantum computer is similar to a random number generator in that its physical mechanism is inaccessible to its users. In this respect, a cloud quantum computer is a black box. In both devices, its users decide the device condition from the output. A framework to achieve this exists in the field of random number generation in the form of statistical tests for random number generators. In the present study, we generated random numbers on a 20-qubit cloud quantum computer and evaluated the condition and stability of its qubits using statistical tests for random number generators. As a res…
A quantum computer that can be constructed based on a superconducting nanocircuits has previously been proposed. We examine the effect of background charge fluctuations on a coupled Josephson charge qubit system used in such a computer. In previous work, the background charge fluctuations were found to be an important dephasing channel for a single Josephson qubit. We investigate the effect of fluctuations in the bias at the charge degeneracy point of a Josephson charge qubit system. Evaluated quantities are gate fidelity and diagonal elements of the qubit's density matrix. The fluctuation lea…
Near-term quantum computers are expected to facilitate material and chemical research through accurate molecular simulations. Several developments have already shown that accurate ground-state energies for small molecules can be evaluated on present-day quantum devices. Although electronically excited states play a vital role in chemical processes and applications, the search for a reliable and practical approach for routine excited-state calculations on near-term quantum devices is ongoing. Inspired by excited-state methods developed for the unitary coupled-cluster theory in quantum chemistry…
The generating function of a Hamiltonian $H$ is defined as $F(t)=\langle e^{-itH}\rangle$, where $t$ is the time and where the expectation value is taken on a given initial quantum state. This function gives access to the different moments of the Hamiltonian $\langle H^{K}\rangle$ at various orders $K$. The real and imaginary parts of $F(t)$ can be respectively evaluated on quantum computers using one extra ancillary qubit with a set of measurement for each value of the time $t$. The low cost in terms of qubits renders it very attractive in the near term period where the number of qubits is li…
Given the Hamiltonian, the evaluation of unitary operators has been at the heart of many quantum algorithms. Motivated by existing deterministic and random methods, we present a hybrid approach, where Hamiltonians with large amplitude are evaluated at each time step, while the remaining terms are evaluated at random. The bound for the mean square error is obtained, together with a concentration bound. The mean square error consists of a variance term and a bias term, arising respectively from the random sampling of the Hamiltonian terms and the operator splitting error. Leveraging on the bias/…
