arXiv subjects

Hartmut Neven

Publications and source records attributed to Hartmut Neven.

At least 73 records · Page 4Linked to original sources

arXiv2018

Universal discriminative quantum neural networks

Quantum mechanics fundamentally forbids deterministic discrimination of quantum states and processes. However, the ability to optimally distinguish various classes of quantum data is an important primitive in quantum information science. In this work, we train near-term quantum circuits to classify data represented by non-orthogonal quantum probability distributions using the Adam stochastic optimization algorithm. This is achieved by iterative interactions of a classical device with a quantum processor to discover the parameters of an unknown non-unitary quantum circuit. This circuit learns to simulates the unknown structure of a generalized quantum measurement, or Positive-Operator-Value-Measure (POVM), that is required to optimally distinguish possible distributions of quantum inputs. Notably we use universal circuit topologies, with a theoretically motivated circuit design, which guarantees that our circuits can in principle learn to perform arbitrary input-output mappings. Our numerical simulations show that shallow quantum circuits could be trained to discriminate among various pure and mixed quantum states exhibiting a trade-off between minimizing erroneous and inconclusive outcomes with comparable performance to theoretically optimal POVMs. We train the circuit on different classes of quantum data and evaluate the generalization error on unseen mixed quantum states. This generalization power hence distinguishes our work from standard circuit optimization and provides an example of quantum machine learning for a task that has inherently no classical analogue.

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arXiv2018

Universal Quantum Control through Deep Reinforcement Learning

Emerging reinforcement learning techniques using deep neural networks have shown great promise in control optimization. They harness non-local regularities of noisy control trajectories and facilitate transfer learning between tasks. To leverage these powerful capabilities for quantum control optimization, we propose a new control framework to simultaneously optimize the speed and fidelity of quantum computation against both leakage and stochastic control errors. For a broad family of two-qubit unitary gates that are important for quantum simulation of many-electron systems, we improve the control robustness by adding control noise into training environments for reinforcement learning agents trained with trusted-region-policy-optimization. The agent control solutions demonstrate a two-order-of-magnitude reduction in average-gate-error over baseline stochastic-gradient-descent solutions and up to a one-order-of-magnitude reduction in gate time from optimal gate synthesis counterparts.

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arXiv2018

Barren plateaus in quantum neural network training landscapes

Many experimental proposals for noisy intermediate scale quantum devices involve training a parameterized quantum circuit with a classical optimization loop. Such hybrid quantum-classical algorithms are popular for applications in quantum simulation, optimization, and machine learning. Due to its simplicity and hardware efficiency, random circuits are often proposed as initial guesses for exploring the space of quantum states. We show that the exponential dimension of Hilbert space and the gradient estimation complexity make this choice unsuitable for hybrid quantum-classical algorithms run on more than a few qubits. Specifically, we show that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits. We argue that this is related to the 2-design characteristic of random circuits, and that solutions to this problem must be studied.

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arXiv2018

Physical qubit calibration on a directed acyclic graph

High-fidelity control of qubits requires precisely tuned control parameters. Typically, these parameters are found through a series of bootstrapped calibration experiments which successively acquire more accurate information about a physical qubit. However, optimal parameters are typically different between devices and can also drift in time, which begets the need for an efficient calibration strategy. Here, we introduce a framework to understand the relationship between calibrations as a directed graph. With this approach, calibration is reduced to a graph traversal problem that is automatable and extensible.

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arXiv2018

Simulation of low-depth quantum circuits as complex undirected graphical models

Near term quantum computers with a high quantity (around 50) and quality (around 0.995 fidelity for two-qubit gates) of qubits will approximately sample from certain probability distributions beyond the capabilities of known classical algorithms on state-of-the-art computers, achieving the first milestone of so-called quantum supremacy. This has stimulated recent progress in classical algorithms to simulate quantum circuits. Classical simulations are also necessary to approximate the fidelity of multiqubit quantum computers using cross entropy benchmarking. Here we present numerical results of a classical simulation algorithm to sample universal random circuits, on a single workstation, with more qubits and depth than previously reported. For example, circuits with $5 \times 9$ qubits of depth 37, $7 \times 8$ qubits of depth 27, and $10 \times (κ> 10)$) qubits of depth 19 are all easy to sample. We also show up to what depth the sampling, or estimation of observables, is trivially parallelizable. The algorithm is related to the "Feynmann path" method to simulate quantum circuits. For low-depth circuits, the algorithm scales exponentially in the depth times the smaller lateral dimension, or the treewidth, as explained in Boixo et. al., and therefore confirms the bounds in that paper. In particular, circuits with $7 \times 7$ qubits and depth 40 remain currently out of reach. Follow up work on a supercomputer environment will tighten this bound.

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arXiv2018

Low Depth Quantum Simulation of Electronic Structure

Quantum simulation of the electronic structure problem is one of the most researched applications of quantum computing. The majority of quantum algorithms for this problem encode the wavefunction using $N$ Gaussian orbitals, leading to Hamiltonians with ${\cal O}(N^4)$ second-quantized terms. We avoid this overhead and extend methods to the condensed phase by utilizing a dual form of the plane wave basis which diagonalizes the potential operator, leading to a Hamiltonian representation with ${\cal O}(N^2)$ second-quantized terms. Using this representation we can implement single Trotter steps of the Hamiltonians with linear gate depth on a planar lattice. Properties of the basis allow us to deploy Trotter and Taylor series based simulations with respective circuit depths of ${\cal O}(N^{7/2})$ and $\widetilde{\cal O}(N^{8/3})$ for fixed charge densities - both are large asymptotic improvements over all prior results. Variational algorithms also require significantly fewer measurements to find the mean energy in this basis, ameliorating a primary challenge of that approach. We conclude with a proposal to simulate the uniform electron gas (jellium) using a low depth variational ansatz realizable on near-term quantum devices. From these results we identify simulations of low density jellium as a promising first setting to explore quantum supremacy in electronic structure.

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arXiv2017

Path-Integral Quantum Monte Carlo simulation with Open-Boundary Conditions

The tunneling decay event of a metastable state in a fully connected quantum spin model can be simulated efficiently by path integral quantum Monte Carlo (QMC) [Isakov $et~al.$, Phys. Rev. Lett. ${\bf 117}$, 180402 (2016).]. This is because the exponential scaling with the number of spins of the thermally-assisted quantum tunneling rate and the Kramers escape rate of QMC are identical [Jiang $et~al.$, Phys. Rev. A ${\bf 95}$, 012322 (2017).], a result of a dominant instantonic tunneling path. In Ref. [1], it was also conjectured that the escape rate in open-boundary QMC is quadratically larger than that of conventional periodic-boundary QMC, therefore, open-boundary QMC might be used as a powerful tool to solve combinatorial optimization problems. The intuition behind this conjecture is that the action of the instanton in open-boundary QMC is a half of that in periodic-boundary QMC. Here, we show that this simple intuition---although very useful in interpreting some numerical results---deviates from the actual situation in several ways. Using a fully connected quantum spin model, we derive a set of conditions on the positions and momenta of the endpoints of the instanton, which remove the extra degrees of freedom due to open boundaries. In comparison, the half-instanton conjecture incorrectly sets the momenta at the endpoints to zero. We also found that the instantons in open-boundary QMC correspond to quantum tunneling events in the symmetric subspace (maximum total angular momentum) at all temperatures, whereas the instantons in periodic-boundary QMC typically lie in subspaces with lower total angular momenta at finite temperatures. This leads to a lesser than quadratic speedup at finite temperatures. We also outline the generalization of the instantonic tunneling method to many-qubit systems without permutation symmetry using spin-coherent-state path integrals.

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arXiv2017

Fourier analysis of sampling from noisy chaotic quantum circuits

Sampling from the output distribution of chaotic quantum evolutions, and of pseudo-random universal quantum circuits in particular, has been proposed as a prominent milestone for near-term quantum supremacy. The same paper notes that chaotic distributions are very sensitive to noise, and under quite general noise models converge to the uniform distribution over bit-strings exponentially in the number of gates. On the one hand, for increasing number of gates, it suffices to choose bit-strings at random to approximate the noisy distribution with fixed statistical distance. On the other hand, cross-entropy benchmarking can be used to gauge the fidelity of an experiment, and the distance to the uniform distribution. We estimate that state-of-the-art classical supercomputers would fail to simulate high-fidelity chaotic quantum circuits with approximately fifty qubits and depth forty. A recent interesting paper proposed a different approximation algorithm to a noisy distribution, extending previous results on the Fourier analysis of commuting quantum circuits. Using the statistical properties of the Porter-Thomas distribution, we show that this new approximation algorithm does not improve random guessing, in polynomial time. Therefore, it confirms previous results and does not represent an additional challenge to the suggested failure stated above.

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arXiv2017

Tunable inductive coupling of superconducting qubits in the strongly nonlinear regime

For a variety of superconducting qubits, tunable interactions are achieved through mutual inductive coupling to a coupler circuit containing a nonlinear Josephson element. In this paper we derive the general interaction mediated by such a circuit under the Born-Oppenheimer Approximation. This interaction naturally decomposes into a classical part, with origin in the classical circuit equations, and a quantum part, associated with the coupler's zero-point energy. Our result is non-perturbative in the qubit-coupler coupling strengths and in the coupler nonlinearity. This can lead to significant departures from previous, linear theories for the inter-qubit coupling, including non-stoquastic and many-body interactions. Our analysis provides explicit and efficiently computable series for any term in the interaction Hamiltonian and can be applied to any superconducting qubit type. We conclude with a numerical investigation of our theory using a case study of two coupled flux qubits, and in particular study the regime of validity of the Born-Oppenheimer Approximation.

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arXiv2017

Characterizing Quantum Supremacy in Near-Term Devices

A critical question for the field of quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of state-of-the-art classical computers, achieving so-called quantum supremacy. We study the task of sampling from the output distributions of (pseudo-)random quantum circuits, a natural task for benchmarking quantum computers. Crucially, sampling this distribution classically requires a direct numerical simulation of the circuit, with computational cost exponential in the number of qubits. This requirement is typical of chaotic systems. We extend previous results in computational complexity to argue more formally that this sampling task must take exponential time in a classical computer. We study the convergence to the chaotic regime using extensive supercomputer simulations, modeling circuits with up to 42 qubits - the largest quantum circuits simulated to date for a computational task that approaches quantum supremacy. We argue that while chaotic states are extremely sensitive to errors, quantum supremacy can be achieved in the near-term with approximately fifty superconducting qubits. We introduce cross entropy as a useful benchmark of quantum circuits which approximates the circuit fidelity. We show that the cross entropy can be efficiently measured when circuit simulations are available. Beyond the classically tractable regime, the cross entropy can be extrapolated and compared with theoretical estimates of circuit fidelity to define a practical quantum supremacy test.

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arXiv2017

Optimizing Variational Quantum Algorithms using Pontryagin's Minimum Principle

We use Pontryagin's minimum principle to optimize variational quantum algorithms. We show that for a fixed computation time, the optimal evolution has a bang-bang (square pulse) form, both for closed and open quantum systems with Markovian decoherence. Our findings support the choice of evolution ansatz in the recently proposed Quantum Approximate Optimization Algorithm. Focusing on the Sherrington-Kirkpatrick spin-glass as an example, we find a system-size independent distribution of the duration of pulses, with characteristic time scale set by the inverse of the coupling constants in the Hamiltonian. The optimality of the bang-bang protocols and the characteristic time scale of the pulses provide an efficient parameterization of the protocol and inform the search for effective hybrid (classical and quantum) schemes for tackling combinatorial optimization problems. For the particular systems we study, we find numerically that the optimal nonadiabatic bang-bang protocols outperform conventional quantum annealing in the presence of weak white additive external noise and weak coupling to a thermal bath modeled with the Redfield master equation.

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arXiv2017

Scaling analysis and instantons for thermally-assisted tunneling and Quantum Monte Carlo simulations

We develop an instantonic calculus to derive an analytical expression for the thermally-assisted tunneling decay rate of a metastable state in a fully connected quantum spin model. The tunneling decay problem can be mapped onto the Kramers escape problem of a classical random dynamical field. This dynamical field is simulated efficiently by path integral Quantum Monte Carlo (QMC). We show analytically that the exponential scaling with the number of spins of the thermally-assisted quantum tunneling rate and the escape rate of the QMC process are identical. We relate this effect to the existence of a dominant instantonic tunneling path. The instanton trajectory is described by nonlinear dynamical mean-field theory equations for a single site magnetization vector, which we solve exactly. Finally, we derive scaling relations for the "spiky" barrier shape when the spin tunnelling and QMC rates scale polynomially with the number of spins $N$ while a purely classical over-the-barrier activation rate scales exponentially with $N$.

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arXiv2016

Artificial quantum thermal bath: Engineering temperature for a many-body quantum system

Temperature determines the relative probability of observing a physical system in an energy state when that system is energetically in equilibrium with its environment. In this paper, we present a theory for engineering the temperature of a quantum system different from its ambient temperature. We define criteria for an engineered quantum bath that, when coupled to a quantum system with Hamiltonian $H$, drives the system to the equilibrium state $\frac{e^{-H/T}}{\rm{Tr}(e^{-H/T})}$ with a tunable parameter $T$. This is basically an analog counterpart of the digital quantum metropolis algorithm. For a system of superconducting qubits, we propose a circuit-QED approximate realization of such an engineered thermal bath consisting of driven lossy resonators. Our proposal opens the path to simulate thermodynamical properties of many-body quantum systems of size not accessible to classical simulations. Also we discuss how an artificial thermal bath can serve as a temperature knob for a hybrid quantum-thermal annealer.

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arXiv2016

What is the Computational Value of Finite Range Tunneling?

Quantum annealing (QA) has been proposed as a quantum enhanced optimization heuristic exploiting tunneling. Here, we demonstrate how finite range tunneling can provide considerable computational advantage. For a crafted problem designed to have tall and narrow energy barriers separating local minima, the D-Wave 2X quantum annealer achieves significant runtime advantages relative to Simulated Annealing (SA). For instances with 945 variables, this results in a time-to-99%-success-probability that is $\sim 10^8$ times faster than SA running on a single processor core. We also compared physical QA with Quantum Monte Carlo (QMC), an algorithm that emulates quantum tunneling on classical processors. We observe a substantial constant overhead against physical QA: D-Wave 2X again runs up to $\sim 10^8$ times faster than an optimized implementation of QMC on a single core. We note that there exist heuristic classical algorithms that can solve most instances of Chimera structured problems in a timescale comparable to the D-Wave 2X. However, we believe that such solvers will become ineffective for the next generation of annealers currently being designed. To investigate whether finite range tunneling will also confer an advantage for problems of practical interest, we conduct numerical studies on binary optimization problems that cannot yet be represented on quantum hardware. For random instances of the number partitioning problem, we find numerically that QMC, as well as other algorithms designed to simulate QA, scale better than SA. We discuss the implications of these findings for the design of next generation quantum annealers.

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arXiv2015

Probabilistic Label Relation Graphs with Ising Models

We consider classification problems in which the label space has structure. A common example is hierarchical label spaces, corresponding to the case where one label subsumes another (e.g., animal subsumes dog). But labels can also be mutually exclusive (e.g., dog vs cat) or unrelated (e.g., furry, carnivore). To jointly model hierarchy and exclusion relations, the notion of a HEX (hierarchy and exclusion) graph was introduced in [7]. This combined a conditional random field (CRF) with a deep neural network (DNN), resulting in state of the art results when applied to visual object classification problems where the training labels were drawn from different levels of the ImageNet hierarchy (e.g., an image might be labeled with the basic level category "dog", rather than the more specific label "husky"). In this paper, we extend the HEX model to allow for soft or probabilistic relations between labels, which is useful when there is uncertainty about the relationship between two labels (e.g., an antelope is "sort of" furry, but not to the same degree as a grizzly bear). We call our new model pHEX, for probabilistic HEX. We show that the pHEX graph can be converted to an Ising model, which allows us to use existing off-the-shelf inference methods (in contrast to the HEX method, which needed specialized inference algorithms). Experimental results show significant improvements in a number of large-scale visual object classification tasks, outperforming the previous HEX model.

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arXiv2015

Systematic Dimensionality Reduction for Quantum Walks: Optimal Spatial Search and Transport on Non-Regular Graphs

Continuous time quantum walks provide an important framework for designing new algorithms and modelling quantum transport and state transfer problems. Often, the graph representing the structure of a problem contains certain symmetries that confine the dynamics to a smaller subspace of the full Hilbert space. In this work, we use invariant subspace methods, that can be computed systematically using Lanczos algorithm, to obtain the reduced set of states that encompass the dynamics of the problem at hand without the specific knowledge of underlying symmetries. First, we apply this method to obtain new instances of graphs where the spatial quantum search algorithm is optimal: complete graphs with broken links and complete bipartite graphs, in particular, the star graph. These examples show that regularity and high-connectivity are not needed to achieve optimal spatial search. We also show that this method considerably simplifies the calculation of quantum transport efficiencies. Furthermore, we observe improved efficiencies by removing a few links from highly symmetric graphs. Finally, we show that this reduction method also allows us to obtain an upper bound for the fidelity of a single qubit transfer on an XY spin network.

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arXiv2015

Understanding Quantum Tunneling through Quantum Monte Carlo Simulations

The tunneling between the two ground states of an Ising ferromagnet is a typical example of many-body tunneling processes between two local minima, as they occur during quantum annealing. Performing quantum Monte Carlo (QMC) simulations we find that the QMC tunneling rate displays the same scaling with system size, as the rate of incoherent tunneling. The scaling in both cases is $O(Δ^2)$, where $Δ$ is the tunneling splitting. An important consequence is that QMC simulations can be used to predict the performance of a quantum annealer for tunneling through a barrier. Furthermore, by using open instead of periodic boundary conditions in imaginary time, equivalent to a projector QMC algorithm, we obtain a quadratic speedup for QMC, and achieve linear scaling in $Δ$. We provide a physical understanding of these results and their range of applicability based on an instanton picture.

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arXiv2015

Totally Corrective Boosting with Cardinality Penalization

We propose a totally corrective boosting algorithm with explicit cardinality regularization. The resulting combinatorial optimization problems are not known to be efficiently solvable with existing classical methods, but emerging quantum optimization technology gives hope for achieving sparser models in practice. In order to demonstrate the utility of our algorithm, we use a distributed classical heuristic optimizer as a stand-in for quantum hardware. Even though this evaluation methodology incurs large time and resource costs on classical computing machinery, it allows us to gauge the potential gains in generalization performance and sparsity of the resulting boosted ensembles. Our experimental results on public data sets commonly used for benchmarking of boosting algorithms decidedly demonstrate the existence of such advantages. If actual quantum optimization were to be used with this algorithm in the future, we would expect equivalent or superior results at much smaller time and energy costs during training. Moreover, studying cardinality-penalized boosting also sheds light on why unregularized boosting algorithms with early stopping often yield better results than their counterparts with explicit convex regularization: Early stopping performs suboptimal cardinality regularization. The results that we present here indicate it is beneficial to explicitly solve the combinatorial problem still left open at early termination.

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