arXiv subjects
Publications and source records attributed to E. Knill.
A few conventions for thinking about and writing quantum pseudocode are proposed. The conventions can be used for presenting any quantum algorithm down to the lowest level and are consistent with a quantum random access machine (QRAM) model for quantum computing. In principle a formal version of quantum pseudocode could be used in a future extension of a conventional language.
Universal control of multiple qubits -- the ability to entangle qubits and to perform arbitrary individual qubit operations -- is a fundamental resource for quantum computation, simulation, and networking. Here, we implement a new laser-free scheme for universal control of trapped ion qubits based on microwave magnetic fields and radiofrequency magnetic field gradients. We demonstrate high-fidelity entanglement and individual control by creating symmetric and antisymmetric two-qubit maximally entangled states with fidelities in the intervals [0.9983, 1] and [0.9964, 0.9988], respectively, at 68% confidence, corrected for state initialization error. This technique is robust against multiple sources of decoherence, usable with essentially any trapped ion species, and has the potential to perform simultaneous entangling operations on many pairs of ions without increasing control signal power or complexity.
A Bell test is a randomized trial that compares experimental observations against the philosophical worldview of local realism. A Bell test requires spatially distributed entanglement, fast and high-efficiency detection and unpredictable measurement settings. Although technology can satisfy the first two of these requirements, the use of physical devices to choose settings in a Bell test involves making assumptions about the physics that one aims to test. Bell himself noted this weakness in using physical setting choices and argued that human `free will' could be used rigorously to ensure unpredictability in Bell tests. Here we report a set of local-realism tests using human choices, which avoids assumptions about predictability in physics. We recruited about 100,000 human participants to play an online video game that incentivizes fast, sustained input of unpredictable selections and illustrates Bell-test methodology. The participants generated 97,347,490 binary choices, which were directed via a scalable web platform to 12 laboratories on five continents, where 13 experiments tested local realism using photons, single atoms, atomic ensembles, and superconducting devices. Over a 12-hour period on 30 November 2016, participants worldwide provided a sustained data flow of over 1,000 bits per second to the experiments, which used different human-generated data to choose each measurement setting. The observed correlations strongly contradict local realism and other realistic positions in bipartite and tripartite scenarios. Project outcomes include closing the `freedom-of-choice loophole' (the possibility that the setting choices are influenced by `hidden variables' to correlate with the particle properties), the utilization of video-game methods for rapid collection of human generated randomness, and the use of networking techniques for global participation in experimental science.
Estimation of quantum states and measurements is crucial for the implementation of quantum information protocols. The standard method for each is quantum tomography. However, quantum tomography suffers from systematic errors caused by imperfect knowledge of the system. We present a procedure to simultaneously characterize quantum states and measurements that mitigates systematic errors by use of a single high-fidelity state preparation and a limited set of high-fidelity unitary operations. Such states and operations are typical of many state-of-the-art systems. For this situation we design a set of experiments and an optimization algorithm that alternates between maximizing the likelihood with respect to the states and measurements to produce estimates of each. In some cases, the procedure does not enable unique estimation of the states. For these cases, we show how one may identify a set of density matrices compatible with the measurements and use a semi-definite program to place bounds on the state's expectation values. We demonstrate the procedure on data from a simulated experiment with two trapped ions.
We describe a quantum limit to measurement of classical spacetimes. Specifically, we formulate a quantum Cramer-Rao lower bound for estimating the single parameter in any one-parameter family of spacetime metrics. We employ the locally covariant formulation of quantum field theory in curved spacetime, which allows for a manifestly background-independent derivation. The result is an uncertainty relation that applies to all globally hyperbolic spacetimes. Among other examples, we apply our method to detection of gravitational waves using the electromagnetic field as a probe, as in laser-interferometric gravitational-wave detectors. Other applications are discussed, from terrestrial gravimetry to cosmology.
We convert propagating qubits encoded as superpositions of zero and one photons to the motion of a micrometer-sized mechanical resonator. Using quantum state tomography, we determine the density matrix of both the propagating photons and the mechanical resonator. By comparing a sufficient set of states before and after conversion, we determine the average process fidelity to be $F_{\textrm{avg}} = 0.83\substack{+0.03-0.06}$ which exceeds the classical bound for the conversion of an arbitrary qubit state. This conversion ability is necessary for using mechanical resonators in emerging quantum communication and modular quantum computation architectures.
We report correlation measurements on two $^9$Be$^+$ ions that violate a chained Bell inequality obeyed by any local-realistic theory. The correlations can be modeled as derived from a mixture of a local-realistic probabilistic distribution and a distribution that violates the inequality. A statistical framework is formulated to quantify the local-realistic fraction allowable in the observed distribution without the fair-sampling or independent-and-identical-distributions assumptions. We exclude models of our experiment whose local-realistic fraction is above 0.327 at the 95 \% confidence level. This bound is significantly lower than 0.586, the minimum fraction derived from a perfect Clauser-Horne-Shimony-Holt inequality experiment. Furthermore, our data provides a device-independent certification of the deterministically created Bell states.
We report high-fidelity laser-beam-induced quantum logic gates on magnetic-field-insensitive qubits comprised of hyperfine states in $^{9}$Be$^+$ ions with a memory coherence time of more than 1 s. We demonstrate single-qubit gates with error per gate of $3.8(1)\times 10^{-5}$. By creating a Bell state with a deterministic two-qubit gate, we deduce a gate error of $8(4)\times10^{-4}$. We characterize the errors in our implementation and discuss methods to further reduce imperfections towards values that are compatible with fault-tolerant processing at realistic overhead.
Entangled states are a crucial resource for quantum-based technologies such as quantum computers and quantum communication systems (1,2). Exploring new methods for entanglement generation is important for diversifying and eventually improving current approaches. Here, we create entanglement in atomic ions by applying laser fields to constrain the evolution to a restricted number of states, in an approach that has become known as "quantum Zeno dynamics" (3-5). With two trapped $^9\rm{Be}^+$ ions, we obtain Bell state fidelities up to $0.990^{+2}_{-5}$, with three ions, a W-state (6) fidelity of $0.910^{+4}_{-7}$ is obtained. Compared to other methods of producing entanglement in trapped ions, this procedure is relatively insensitive to certain imperfections such as fluctuations in laser intensity, laser frequency, and ion-motion frequencies.
We apply a distance-based Bell-test analysis method [E. Knill et al., Phys. Rev. A. 91, 032105 (2015)] to three experimental data sets where conventional analyses failed or required additional assumptions. The first is produced from a new classical source exploiting a "coincidence-time loophole" for which standard analysis falsely shows a Bell violation. The second is from a source previously shown to violate a Bell inequality; the distance-based analysis agrees with the previous results but with fewer assumptions. The third data set does not show a violation with standard analysis despite the high source quality, but is shown to have a strong violation with the distance-based analysis method.
As experimental platforms for quantum information processing continue to mature, characterization of the quality of unitary gates that can be applied to their quantum bits (qubits) becomes essential. Eventually, the quality must be sufficiently high to support arbitrarily long quantum computations. Randomized benchmarking already provides a platform-independent method for assessing the quality of one-qubit rotations. Here we describe an extension of this method to multi-qubit gates. We provide a platform-independent protocol for evaluating the performance of experimental Clifford unitaries, which form the basis of fault-tolerant quantum computing. We implemented the benchmarking protocol with trapped-ion two-qubit phase gates and one-qubit gates and found an error per random two-qubit Clifford unitary of $0.162 \pm 0.008$, thus setting the first benchmark for such unitaries. By implementing a second set of sequences with an extra two-qubit phase gate at each step, we extracted an error per phase gate of $0.069 \pm 0.017$. We conducted these experiments with movable, sympathetically cooled ions in a multi-zone Paul trap - a system that can in principle be scaled to larger numbers of ions.
We exploit a new theory of duality transformations to construct dual representations of models incompatible with traditional duality transformations. Hence we obtain a solution to the long-standing problem of non-Abelian dualities that hinges on two key observations: (i) from the point of view of dualities, whether the group of symmetries of a model is or is not Abelian is unimportant, and (ii) the new theory of dualities that we exploit includes traditional duality transformations, but also introduces in a natural way more general transformations.
With a 9Be+ trapped-ion hyperfine-states qubit, we demonstrate an error probability per randomized single-qubit gate of 2.0(2) x 10^-5, below the threshold estimate of 10^-4 commonly considered sufficient for fault-tolerant quantum computing. The 9Be+ ion is trapped above a microfabricated surface-electrode ion trap and is manipulated with microwaves applied to a trap electrode. The achievement of low single-qubit-gate errors is an essential step toward the construction of a scalable quantum computer.
We perform state tomography of an itinerant squeezed state of the microwave field prepared by a Josephson parametric amplifier (JPA). We use a second JPA as a pre-amplifier to improve the quantum efficiency of the field quadrature measurement (QM) from 2% to 36 +/- 4%. Without correcting for the detection inefficiency we observe a minimum quadrature variance which is 69 +/- 8% of the variance of the vacuum. We reconstruct the state's density matrix by a maximum likelihood method and infer that the squeezed state has a minimum variance less than 40% of the vacuum, with uncertainty mostly caused by calibration systematics.
Consider a path of non-degenerate eigenstates of unitary operators or Hamiltonians with minimum eigenvalue gap G. The eigenpath traversal problem is to transform one or more copies of the initial to the final eigenstate. Solutions to this problem have applications ranging from quantum physics simulation to optimization. For Hamiltonians, the conventional way of doing this is by applying the adiabatic theorem. We give ``digital'' methods for performing the transformation that require no assumption on path continuity or differentiability other than the absence of large jumps. Given sufficient information about eigenvalues and overlaps between states on the path, the transformation can be accomplished with complexity O(L/G log(L/e)), where L is the angular length of the path and e is a specified bound on the error of the output state. We show that the required information can be obtained in a first set of transformations, whose complexity per state transformed has an additional factor that depends logarithmically on a maximum angular velocity along the path. This velocity is averaged over constant angular distances and does not require continuity. Our methods have substantially better behavior than conventional adiabatic algorithms, with fewer conditions on the path. They also improve on the previously best digital methods and demonstrate that path length and the gap are the primary parameters that determine the complexity of state transformation along a path.
A computation in adiabatic quantum computing is implemented by traversing a path of nondegenerate eigenstates of a continuous family of Hamiltonians. We introduce a method that traverses a discretized form of the path: At each step we apply the instantaneous Hamiltonian for a random time. The resulting decoherence approximates a projective measurement onto the desired eigenstate, achieving a version of the quantum Zeno effect. If negative evolution times can be implemented with constant overhead, then the average absolute evolution time required by our method is O(L^2/Delta) for constant error probability, where L is the length of the path of eigenstates and Delta is the minimum spectral gap of the Hamiltonian. Making explicit the dependence on the path length is useful for cases where L does not depend on Delta. The complexity of our method has a logarithmic improvement over previous algorithms of this type. The same cost applies to the discrete-time case, where a family of unitary operators is given and each unitary and its inverse can be used. Restriction to positive evolution times incurs an error that decreases exponentially with the cost. Applications of this method to unstructured search and quantum sampling are considered. In particular, we discuss the quantum simulated annealing algorithm for solving combinatorial optimization problems. This algorithm provides a quadratic speed-up in the gap of the stochastic matrix over its classical counterpart implemented via Markov chain Monte Carlo.
We describe a quantum algorithm that solves combinatorial optimization problems by quantum simulation of a classical simulated annealing process. Our algorithm exploits quantum walks and the quantum Zeno effect induced by evolution randomization. It requires order $1/\sqrtδ$ steps to find an optimal solution with bounded error probability, where $δ$ is the minimum spectral gap of the stochastic matrices used in the classical annealing process. This is a quadratic improvement over the order $1/δ$ steps required by the latter.
In building a quantum information processor (QIP), the challenge is to coherently control a large quantum system well enough to perform an arbitrary quantum algorithm and to be able to correct errors induced by decoherence. Nuclear magnetic resonance (NMR) QIPs offer an excellent test-bed on which to develop and benchmark tools and techniques to control quantum systems. Two main issues to consider when designing control methods are accuracy and efficiency, for which two complementary approaches have been developed so far to control qubit registers with liquid-state NMR methods. The first applies optimal control theory to numerically optimize the control fields to implement unitary operations on low dimensional systems with high fidelity. The second technique is based on the efficient optimization of a sequence of imperfect control elements so that implementation of a full quantum algorithm is possible while minimizing error accumulation. This article summarizes our work in implementing both of these methods. Furthermore, we show that taken together, they form a basis to design quantum-control methods for a block-architecture QIP so that large system size is not a barrier to implementing optimal control techniques.