arXiv subjects

Patrick J. Coles

Publications and source records attributed to Patrick J. Coles.

At least 73 records · Page 4Linked to original sources

arXiv2021

Variational Quantum Algorithms

Applications such as simulating complicated quantum systems or solving large-scale linear algebra problems are very challenging for classical computers due to the extremely high computational cost. Quantum computers promise a solution, although fault-tolerant quantum computers will likely not be available in the near future. Current quantum devices have serious constraints, including limited numbers of qubits and noise processes that limit circuit depth. Variational Quantum Algorithms (VQAs), which use a classical optimizer to train a parametrized quantum circuit, have emerged as a leading strategy to address these constraints. VQAs have now been proposed for essentially all applications that researchers have envisioned for quantum computers, and they appear to the best hope for obtaining quantum advantage. Nevertheless, challenges remain including the trainability, accuracy, and efficiency of VQAs. Here we overview the field of VQAs, discuss strategies to overcome their challenges, and highlight the exciting prospects for using them to obtain quantum advantage.

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arXiv2021

Effect of barren plateaus on gradient-free optimization

Barren plateau landscapes correspond to gradients that vanish exponentially in the number of qubits. Such landscapes have been demonstrated for variational quantum algorithms and quantum neural networks with either deep circuits or global cost functions. For obvious reasons, it is expected that gradient-based optimizers will be significantly affected by barren plateaus. However, whether or not gradient-free optimizers are impacted is a topic of debate, with some arguing that gradient-free approaches are unaffected by barren plateaus. Here we show that, indeed, gradient-free optimizers do not solve the barren plateau problem. Our main result proves that cost function differences, which are the basis for making decisions in a gradient-free optimization, are exponentially suppressed in a barren plateau. Hence, without exponential precision, gradient-free optimizers will not make progress in the optimization. We numerically confirm this by training in a barren plateau with several gradient-free optimizers (Nelder-Mead, Powell, and COBYLA algorithms), and show that the numbers of shots required in the optimization grows exponentially with the number of qubits.

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arXiv2021

Theory of overparametrization in quantum neural networks

The prospect of achieving quantum advantage with Quantum Neural Networks (QNNs) is exciting. Understanding how QNN properties (e.g., the number of parameters $M$) affect the loss landscape is crucial to the design of scalable QNN architectures. Here, we rigorously analyze the overparametrization phenomenon in QNNs with periodic structure. We define overparametrization as the regime where the QNN has more than a critical number of parameters $M_c$ that allows it to explore all relevant directions in state space. Our main results show that the dimension of the Lie algebra obtained from the generators of the QNN is an upper bound for $M_c$, and for the maximal rank that the quantum Fisher information and Hessian matrices can reach. Underparametrized QNNs have spurious local minima in the loss landscape that start disappearing when $M\geq M_c$. Thus, the overparametrization onset corresponds to a computational phase transition where the QNN trainability is greatly improved by a more favorable landscape. We then connect the notion of overparametrization to the QNN capacity, so that when a QNN is overparametrized, its capacity achieves its maximum possible value. We run numerical simulations for eigensolver, compilation, and autoencoding applications to showcase the overparametrization computational phase transition. We note that our results also apply to variational quantum algorithms and quantum optimal control.

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arXiv2021

Adaptive shot allocation for fast convergence in variational quantum algorithms

Variational Quantum Algorithms (VQAs) are a promising approach for practical applications like chemistry and materials science on near-term quantum computers as they typically reduce quantum resource requirements. However, in order to implement VQAs, an efficient classical optimization strategy is required. Here we present a new stochastic gradient descent method using an adaptive number of shots at each step, called the global Coupled Adaptive Number of Shots (gCANS) method, which improves on prior art in both the number of iterations as well as the number of shots required. These improvements reduce both the time and money required to run VQAs on current cloud platforms. We analytically prove that in a convex setting gCANS achieves geometric convergence to the optimum. Further, we numerically investigate the performance of gCANS on some chemical configuration problems. We also consider finding the ground state for an Ising model with different numbers of spins to examine the scaling of the method. We find that for these problems, gCANS compares favorably to all of the other optimizers we consider.

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arXiv2021

Unified approach to data-driven quantum error mitigation

Achieving near-term quantum advantage will require effective methods for mitigating hardware noise. Data-driven approaches to error mitigation are promising, with popular examples including zero-noise extrapolation (ZNE) and Clifford data regression (CDR). Here we propose a novel, scalable error mitigation method that conceptually unifies ZNE and CDR. Our approach, called variable-noise Clifford data regression (vnCDR), significantly outperforms these individual methods in numerical benchmarks. vnCDR generates training data first via near-Clifford circuits (which are classically simulable) and second by varying the noise levels in these circuits. We employ a noise model obtained from IBM's Ourense quantum computer to benchmark our method. For the problem of estimating the energy of an 8-qubit Ising model system, vnCDR improves the absolute energy error by a factor of 33 over the unmitigated results and by factors 20 and 1.8 over ZNE and CDR, respectively. For the problem of correcting observables from random quantum circuits with 64 qubits, vnCDR improves the error by factors of 2.7 and 1.5 over ZNE and CDR, respectively.

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arXiv2021

Long-time simulations with high fidelity on quantum hardware

Moderate-size quantum computers are now publicly accessible over the cloud, opening the exciting possibility of performing dynamical simulations of quantum systems. However, while rapidly improving, these devices have short coherence times, limiting the depth of algorithms that may be successfully implemented. Here we demonstrate that, despite these limitations, it is possible to implement long-time, high fidelity simulations on current hardware. Specifically, we simulate an XY-model spin chain on the Rigetti and IBM quantum computers, maintaining a fidelity of at least 0.9 for over 600 time steps. This is a factor of 150 longer than is possible using the iterated Trotter method. Our simulations are performed using a new algorithm that we call the fixed state Variational Fast Forwarding (fsVFF) algorithm. This algorithm decreases the circuit depth and width required for a quantum simulation by finding an approximate diagonalization of a short time evolution unitary. Crucially, fsVFF only requires finding a diagonalization on the subspace spanned by the initial state, rather than on the total Hilbert space as with previous methods, substantially reducing the required resources. We further demonstrate the viability of fsVFF through large numerical implementations of the algorithm, as well as an analysis of its noise resilience and the scaling of simulation errors.

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arXiv2021

Quantum simulation of operator spreading in the chaotic Ising model

There is great interest in using near-term quantum computers to simulate and study foundational problems in quantum mechanics and quantum information science, such as the scrambling measured by an out-of-time-ordered correlator (OTOC). Here we use an IBM Q processor, quantum error mitigation, and weaved Trotter simulation to study high-resolution operator spreading in a 4-spin Ising model as a function of space, time, and integrability. Reaching 4 spins while retaining high circuit fidelity is made possible by the use of a physically motivated fixed-node variant of the OTOC, allowing scrambling to be estimated without overhead. We find clear signatures of ballistic operator spreading in a chaotic regime, as well as operator localization in an integrable regime. The techniques developed and demonstrated here open up the possibility of using cloud-based quantum computers to study and visualize scrambling phenomena, as well as quantum information dynamics more generally.

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arXiv2021

Sub-Quantum Fisher Information

The Quantum Fisher Information (QFI) plays a crucial role in quantum information theory and in many practical applications such as quantum metrology. However, computing the QFI is generally a computationally demanding task. In this work we analyze a lower bound on the QFI which we call the sub-Quantum Fisher Information (sub-QFI). The bound can be efficiently estimated on a quantum computer for an $n$-qubit state using $2n$ qubits. The sub-QFI is based on the super-fidelity, an upper bound on Uhlmann's fidelity. We analyze the sub-QFI in the context of unitary families, where we derive several crucial properties including its geometrical interpretation. In particular, we prove that the QFI and the sub-QFI are maximized for the same optimal state, which implies that the sub-QFI is faithful to the QFI in the sense that both quantities share the same global extrema. Based on this faithfulness, the sub-QFI acts as an efficiently computable surrogate for the QFI for quantum sensing and quantum metrology applications. Finally, we provide additional meaning to the sub-QFI as a measure of coherence, asymmetry, and purity loss.

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arXiv2021

Higher Order Derivatives of Quantum Neural Networks with Barren Plateaus

Quantum neural networks (QNNs) offer a powerful paradigm for programming near-term quantum computers and have the potential to speedup applications ranging from data science to chemistry to materials science. However, a possible obstacle to realizing that speedup is the Barren Plateau (BP) phenomenon, whereby the gradient vanishes exponentially in the system size $n$ for certain QNN architectures. The question of whether high-order derivative information such as the Hessian could help escape a BP was recently posed in the literature. Here we show that the elements of the Hessian are exponentially suppressed in a BP, so estimating the Hessian in this situation would require a precision that scales exponentially with $n$. Hence, Hessian-based approaches do not circumvent the exponential scaling associated with BPs. We also show the exponential suppression of higher order derivatives. Hence, BPs will impact optimization strategies that go beyond (first-order) gradient descent. In deriving our results, we prove novel, general formulas that can be used to analytically evaluate any high-order partial derivative on quantum hardware. These formulas will likely have independent interest and use for training quantum neural networks (outside of the context of BPs).

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arXiv2021

Experimental Quantum Learning of a Spectral Decomposition

Currently available quantum hardware allows for small scale implementations of quantum machine learning algorithms. Such experiments aid the search for applications of quantum computers by benchmarking the near-term feasibility of candidate algorithms. Here we demonstrate the quantum learning of a two-qubit unitary by a sequence of three parameterized quantum circuits containing a total of 21 variational parameters. Moreover, we variationally diagonalize the unitary to learn its spectral decomposition, i.e., its eigenvalues and eigenvectors. We illustrate how this can be used as a subroutine to compress the depth of dynamical quantum simulations. One can view our implementation as a demonstration of entanglement-enhanced machine learning, as only a single (entangled) training data pair is required to learn a 4x4 unitary matrix.

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arXiv2021

Qubit-efficient exponential suppression of errors

Achieving a practical advantage with near-term quantum computers hinges on having effective methods to suppress errors. Recent breakthroughs have introduced methods capable of exponentially suppressing errors by preparing multiple noisy copies of a state and virtually distilling a more purified version. Here we present an alternative method, the Resource-Efficient Quantum Error Suppression Technique (REQUEST), that adapts this breakthrough to much fewer qubits by making use of active qubit resets, a feature now available on commercial platforms. Our approach exploits a space/time trade-off to achieve a similar error reduction using only $2N+1$ qubits as opposed to $MN+1$ qubits, for $M$ copies of an $N$ qubit state. Additionally, we propose a method using near-Clifford circuits to find the optimal number of these copies in the presence of realistic noise, which limits this error suppression. We perform a numerical comparison between the original method and our qubit-efficient version with a realistic trapped-ion noise model. We find that REQUEST can reproduce the exponential suppression of errors of the virtual distillation approach, while out-performing virtual distillation when fewer than $3N+1$ qubits are available. Finally, we examine the scaling of the number of shots $N_S$ required for REQUEST to achieve useful corrections. We find that $N_S$ remains reasonable well into the quantum advantage regime where $N$ is hundreds of qubits.

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arXiv2021

Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits

Variational quantum algorithms (VQAs) optimize the parameters $\vecθ$ of a parametrized quantum circuit $V(\vecθ)$ to minimize a cost function $C$. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming $V(\vecθ)$ is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining $C$ in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when $V(\vecθ)$ is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining $C$ with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of $V(\vecθ)$ is $\mathcal{O}(\log n)$. Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.

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arXiv2021

Machine learning of noise-resilient quantum circuits

Noise mitigation and reduction will be crucial for obtaining useful answers from near-term quantum computers. In this work, we present a general framework based on machine learning for reducing the impact of quantum hardware noise on quantum circuits. Our method, called noise-aware circuit learning (NACL), applies to circuits designed to compute a unitary transformation, prepare a set of quantum states, or estimate an observable of a many-qubit state. Given a task and a device model that captures information about the noise and connectivity of qubits in a device, NACL outputs an optimized circuit to accomplish this task in the presence of noise. It does so by minimizing a task-specific cost function over circuit depths and circuit structures. To demonstrate NACL, we construct circuits resilient to a fine-grained noise model derived from gate set tomography on a superconducting-circuit quantum device, for applications including quantum state overlap, quantum Fourier transform, and W-state preparation.

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arXiv2021

Security proof of practical quantum key distribution with detection-efficiency mismatch

Quantum key distribution (QKD) protocols with threshold detectors are driving high-performance QKD demonstrations. The corresponding security proofs usually assume that all physical detectors have the same detection efficiency. However, the efficiencies of the detectors used in practice might show a mismatch depending on the manufacturing and setup of these detectors. A mismatch can also be induced as the different spatial-temporal modes of an incoming signal might couple differently to a detector. Here we develop a method that allows to provide security proofs without the usual assumption. Our method can take the detection-efficiency mismatch into account without having to restrict the attack strategy of the adversary. Especially, we do not rely on any photon-number cut-off of incoming signals such that our security proof is directly applicable to practical situations. We illustrate our method for a receiver that is designed for polarization encoding and is sensitive to a number of spatial-temporal modes. In our detector model, the absence of quantum interference between any pair of spatial-temporal modes is assumed. For a QKD protocol with this detector model, we can perform a security proof with characterized efficiency mismatch and without photon-number cut-off assumption. Our method also shows that in the absence of efficiency mismatch in our detector model, the key rate increases if the loss due to detection inefficiency is assumed to be outside of the adversary's control, as compared to the view where for a security proof this loss is attributed to the action of the adversary.

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arXiv2021

Large gradients via correlation in random parameterized quantum circuits

Scaling of variational quantum algorithms to large problem sizes requires efficient optimization of random parameterized quantum circuits. For such circuits with uncorrelated parameters, the presence of exponentially vanishing gradients in cost function landscapes is an obstacle to optimization by gradient descent methods. In this work, we prove that reducing the dimensionality of the parameter space by utilizing circuit modules containing spatially or temporally correlated gate layers can allow one to circumvent the vanishing gradient phenomenon. Examples are drawn from random separable circuits and asymptotically optimal variational versions of Grover's algorithm based on the quantum alternating operator ansatz (QAOA). In the latter scenario, our bounds on cost function variation imply a transition between vanishing gradients and efficient trainability as the number of layers is increased toward $\mathcal{O}(2^{n/2})$, the optimal oracle complexity of quantum unstructured search.

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arXiv2020

Variational Fast Forwarding for Quantum Simulation Beyond the Coherence Time

Trotterization-based, iterative approaches to quantum simulation are restricted to simulation times less than the coherence time of the quantum computer, which limits their utility in the near term. Here, we present a hybrid quantum-classical algorithm, called Variational Fast Forwarding (VFF), for decreasing the quantum circuit depth of quantum simulations. VFF seeks an approximate diagonalization of a short-time simulation to enable longer-time simulations using a constant number of gates. Our error analysis provides two results: (1) the simulation error of VFF scales at worst linearly in the fast-forwarded simulation time, and (2) our cost function's operational meaning as an upper bound on average-case simulation error provides a natural termination condition for VFF. We implement VFF for the Hubbard, Ising, and Heisenberg models on a simulator. Additionally, we implement VFF on Rigetti's quantum computer to demonstrate simulation beyond the coherence time. Finally, we show how to estimate energy eigenvalues using VFF.

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arXiv2020

Correlation-Informed Permutation of Qubits for Reducing Ansatz Depth in VQE

The Variational Quantum Eigensolver (VQE) is a method of choice to solve the electronic structure problem for molecules on near-term gate-based quantum computers. However, the circuit depth is expected to grow significantly with problem size. Increased depth can both degrade the accuracy of the results and reduce trainability. In this work, we propose a novel approach to reduce ansatz circuit depth. Our approach, called PermVQE, adds an additional optimization loop to VQE that permutes qubits in order to solve for the qubit Hamiltonian that minimizes long-range correlations in the ground state. The choice of permutations is based on mutual information, which is a measure of interaction between electrons in spin-orbitals. Encoding strongly interacting spin-orbitals into proximal qubits on a quantum chip naturally reduces the circuit depth needed to prepare the ground state. For representative molecular systems, LiH, H$_2$, (H$_2$)$_2$, H$_4$, and H$_3^+$, we demonstrate for linear qubit connectivity that placing entangled qubits in close proximity leads to shallower depth circuits required to reach a given eigenvalue-eigenvector accuracy. This approach can be extended to any qubit connectivity and can significantly reduce the depth required to reach a desired accuracy in VQE. Moreover, our approach can be applied to other variational quantum algorithms beyond VQE.

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arXiv2020

Variational Hamiltonian Diagonalization for Dynamical Quantum Simulation

Dynamical quantum simulation may be one of the first applications to see quantum advantage. However, the circuit depth of standard Trotterization methods can rapidly exceed the coherence time of noisy quantum computers. This has led to recent proposals for variational approaches to dynamical simulation. In this work, we aim to make variational dynamical simulation even more practical and near-term. We propose a new algorithm called Variational Hamiltonian Diagonalization (VHD), which approximately transforms a given Hamiltonian into a diagonal form that can be easily exponentiated. VHD allows for fast forwarding, i.e., simulation beyond the coherence time of the quantum computer with a fixed-depth quantum circuit. It also removes Trotterization error and allows simulation of the entire Hilbert space. We prove an operational meaning for the VHD cost function in terms of the average simulation fidelity. Moreover, we prove that the VHD cost function does not exhibit a shallow-depth barren plateau, i.e., its gradient does not vanish exponentially. Our proof relies on locality of the Hamiltonian, and hence we connect locality to trainability. Our numerical simulations verify that VHD can be used for fast-forwarding dynamics.

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