Search

We describe an efficient quantum algorithm for the quantum Schur transform. The Schur transform is an operation on a quantum computer that maps the standard computational basis to a basis composed of irreducible representations of the unitary and symmetric groups. We simplify and extend the algorithm of Bacon, Chuang, and Harrow, and provide a new practical construction as well as sharp theoretical and practical analyses. Our algorithm decomposes the Schur transform on $n$ qubits into $O\left(n^4\log\left(\frac{n}{\epsilon}\right)\right)$ operators in the Clifford+T fault-tolerant gate set and uses exactly $2\lfloor\log_2(n)\rfloor-1$ ancillary qubits. We extend our qubit algorithm to decompose the Schur transform on $n$ qudits of dimension $d$ into $O\left(d^{1+p}n^{3d}\log^p\left(\frac{d n}{\epsilon}\right)\right)$ primitive operators from any universal gate set, for $p\approx3.97$., Comment: Erratum: the algorithm in this paper implements the inverse Schur transform. Thanks to Adam Wills for pointing this out! Since the algorithm compiles the inverse Schur transform into a quantum circuit, the Schur transform can be obtained by inverting that circuit, so the costs of the algorithm are unchanged. See paper for details

This Resource Letter surveys the history and modern developments in the field of quantum information. It is written to guide advanced undergraduates, beginning graduate students, and other new researchers to the theoretical and experimental aspects of this field. The topics covered include quantum states and processes, quantum coding and cryptography, quantum computation, the experimental implementation of quantum information processing, and the role of quantum information in the fundamental properties and foundations of physical theories.

We describe how single Josephson junctions can be connected together capacitively to form a two-qubit system. We discuss the general behavior of this system show the energy level dependence on various junction parameters and choice of coupling strengths. We also discuss measurement techniques for reading out both qubits which are relevant to our ongoing experiments.

We review briefly the problems that are driving the search for a quantum computer. These include, primarily, methods for encryption and decryption based on Shor’s algorithm for factoring large integers and the use of Pell’s equation for encryption. We also outline some of the approaches that have been suggested for implementing a quantum computer and then focus on Josephson-junction systems as qubits. We have been investigating the current-biased Josephson junction for this application, a suggestion we made about 2 years ago. We have studied macroscopic quantum tunneling and energy level spectroscopy, using microwaves, in single junctions and recently we have begun measurements of the two-quantum bit (qubit) system, i.e. two capacitively coupled junctions. Theoretical studies of energy levels and their dynamic evolution are also in progress. In the present report we discuss the basics of single Josephson junctions and compare their potential as qubits with the potentials of other systems. We also discuss our future plans to obtain greater isolation of the junctions from sources of decoherence and to develop realistic qubits. An important first step must be to exhibit quantum entanglement and measure coherence times. Then it must be shown that the states of the qubits can be initialized, that gate operations can be performed, and that the results can be read out.

Quantum walks have been shown to have impressive transport properties compared to classical random walks. However, imperfections in the quantum walk algorithm can destroy any quantum mechanical speed-up due to Anderson localization. We numerically study the effect of static disorder on a quantum walk on the glued trees graph. For small disorder, we find that the dominant effect is a type of quantum decay, and not quantum localization. For intermediate disorder, there is a crossover to diffusive transport, while a localization transition is observed at large disorder, in agreement with Anderson localization on the Cayley tree., Comment: 12 pages, 13 figures

An all-resonant method is proposed to control the quantum state of superconducting resonators. This approach uses a tunable artificial atom linearly coupled to resonators, and allows for efficient routes to Fock state synthesis, qudit logic operations, and synthesis of NOON states. This resonant approach is theoretically analyzed, and found to perform signficantly better than existing proposals using the same technology., 4 + epsilon pages, plus supplemental material

We study quantum information processing using superpositions of Fock states in superconducting resonators, as quantum $d$-level systems (qudits). A universal set of single and coupled logic gates is theoretically proposed for resonators coupled by superconducting circuits of Josephson juctions. These gates use experimentally demonstrated interactions, and provide an attractive route to quantum information processing using harmonic oscillator modes., 10 pages, 14 figures, decoherence calculations added

We present a theoretical analysis of methods to synthesize entangled states of two superconducting resonators. These methods use experimentally demonstrated interactions of resonators with artificial atoms, and offer efficient routes to generate nonclassical states. We analyze physical implementations, energy level structure, and the effects of decoherence through detailed dynamical simulations., 14 pages, 10 figures

We study the routing of quantum information in parallel on multi-dimensional networks of tunable qubits and oscillators. These theoretical models are inspired by recent experiments in superconducting circuits using Josephson junctions and resonators. We show that perfect parallel state transfer is possible for certain networks of harmonic oscillator modes. We further extend this to the distribution of entanglement between every pair of nodes in the network, finding that the routing efficiency of hypercube networks is both optimal and robust in the presence of dissipation and finite bandwidth., 5 pages, 3 figures

We present a method to synthesize an arbitrary quantum state of two superconducting resonators. This state-synthesis algorithm utilizes a coherent interaction of each resonator with a tunable artificial atom to create entangled quantum superpositions of photon number (Fock) states in the resonators. We theoretically analyze this approach, showing that it can efficiently synthesize NOON states, with large photon numbers, using existing technology., 4 + epsilon pages, 3 figures, 1 table, fixed several factors of two, results unchanged

Superconducting quantum circuits, such as the superconducting phase qubit, have multiple quantum states that can interfere with ideal qubit operation. The use of multiple frequency control pulses, resonant with the energy differences of the multi-state system, is theoretically explored. An analytical method to design such control pulses is developed, using a generalization of the Floquet method to multiple frequency controls. This method is applicable to optimizing the control of both superconducting qubits and qudits, and is found to be in excellent agreement with time-dependent numerical simulations., 11 pages, 6 figures

We analyze the behavior of a dc Superconducting Quantum Interference Device (SQUID) phase qubit in which one junction acts as a phase qubit and the rest of the device provides isolation from dissipation and noise in the bias leads. Ignoring dissipation, we find the two-dimensional Hamiltonian of the system and use numerical methods and a cubic approximation to solve Schrodinger's equation for the eigenstates, energy levels, tunneling rates, and expectation value of the currents in the junctions. Using these results, we investigate how well this design provides isolation while preserving the characteristics of a phase qubit. In addition, we show that the expectation value of current flowing through the isolation junction depends on the state of the qubit and can be used for non-destructive read out of the qubit state., 38 pages, 7 figures

Rabi oscillations have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (qubits) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase qubit (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi frequency, and two-photon Rabi frequency are compared to measurements made on a dc SQUID phase qubit with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment., 4 pages, 4 figures, accepted for publication in IEEE Trans. Appl. Supercond

We study the quantum mechanical behavior of a macroscopic, three-body, superconducting circuit. Microwave spectroscopy on our system, a resonator coupling two large Josephson junctions, produced complex energy spectra well explained by quantum theory over a large frequency range. By tuning each junction separately into resonance with the resonator, we first observe strong coupling between each junction and the resonator. Bringing both junctions together into resonance with the resonator, we find spectroscopic evidence for entanglement between all three degrees of freedom and suggest a new method for controllable coupling of distant qubits, a key step toward quantum computation., Comment: 4 pages, 3 figures

Based on a quantum analysis of two capacitively coupled current-biased Josephson junctions, we propose two fundamental two-qubit quantum logic gates. Each of these gates, when supplemented by single-qubit operations, is sufficient for universal quantum computation. Numerical solutions of the time-dependent Schroedinger equation demonstrate that these operations can be performed with good fidelity., Comment: 4 pages, 5 figures, revised for publication

We present spectroscopic evidence for the creation of entangled macroscopic quantum states in two current-biased Josephson-junction qubits coupled by a capacitor. The individual junction bias currents are used to control the interaction between the qubits by tuning the energy level spacings of the junctions in and out of resonance with each other. Microwave spectroscopy in the 4 to 6 gigahertzrange at 20 millikelvin reveals energy levels that agree well with theoretical results for entangled states. The single qubits are spatially separate, and the entangled states extend over the 0.7-millimeter distance between the two qubits.

Practical quantum computation will require a scalable, robust system to generate and process information with precise control. This is now possible using a superconducting circuit and a little quantum magic.