arXiv subjects

Kunal Sharma

Publications and source records attributed to Kunal Sharma.

At least 37 records · Page 2Linked to original sources

arXiv2024

Adiabatic Limit of Calderon Projector on Manifold with Cylindrical End

For a Riemannian manifold with a cylindrical end, consider a Dirac-type operator that is asymptotically product type with the generalized Atiyah-Patodi-Singer boundary condition on any finite portion of the cylinder. In the present work we consider the problem of constructing the Calderon projector in this setting and studying the adiabatic limit of it along the cylindrical end. As a consequence, we extend a result of Nicolaescu on adiabatic limits of Cauchy data spaces. The proof leverages resolvent and its estimates in the framework of the b-calculus needed for the construction of the Calderon projector corresponding to our Dirac-type operator.

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arXiv2023

Estimating distinguishability measures on quantum computers

The performance of a quantum information processing protocol is ultimately judged by distinguishability measures that quantify how distinguishable the actual result of the protocol is from the ideal case. The most prominent distinguishability measures are those based on the fidelity and trace distance, due to their physical interpretations. In this paper, we propose and review several algorithms for estimating distinguishability measures based on trace distance and fidelity. The algorithms can be used for distinguishing quantum states, channels, and strategies (the last also known in the literature as "quantum combs"). The fidelity-based algorithms offer novel physical interpretations of these distinguishability measures in terms of the maximum probability with which a single prover (or competing provers) can convince a verifier to accept the outcome of an associated computation. We simulate many of these algorithms by using a variational approach with parameterized quantum circuits. We find that the simulations converge well in both the noiseless and noisy scenarios, for all examples considered. Furthermore, the noisy simulations exhibit a parameter noise resilience. Finally, we establish a strong relationship between various quantum computational complexity classes and distance estimation problems.

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arXiv2023

Effect of non-unital noise on random circuit sampling

In this work, drawing inspiration from the type of noise present in real hardware, we study the output distribution of random quantum circuits under practical non-unital noise sources with constant noise rates. We show that even in the presence of unital sources like the depolarizing channel, the distribution, under the combined noise channel, never resembles a maximally entropic distribution at any depth. To show this, we prove that the output distribution of such circuits never anticoncentrates $\unicode{x2014}$ meaning it is never too "flat" $\unicode{x2014}$ regardless of the depth of the circuit. This is in stark contrast to the behavior of noiseless random quantum circuits or those with only unital noise, both of which anticoncentrate at sufficiently large depths. As consequences, our results have interesting algorithmic implications on both the hardness and easiness of noisy random circuit sampling, since anticoncentration is a critical property exploited by both state-of-the-art classical hardness and easiness results.

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arXiv2023

Analytic theory for the dynamics of wide quantum neural networks

Parameterized quantum circuits can be used as quantum neural networks and have the potential to outperform their classical counterparts when trained for addressing learning problems. To date, much of the results on their performance on practical problems are heuristic in nature. In particular, the convergence rate for the training of quantum neural networks is not fully understood. Here, we analyze the dynamics of gradient descent for the training error of a class of variational quantum machine learning models. We define wide quantum neural networks as parameterized quantum circuits in the limit of a large number of qubits and variational parameters. We then find a simple analytic formula that captures the average behavior of their loss function and discuss the consequences of our findings. For example, for random quantum circuits, we predict and characterize an exponential decay of the residual training error as a function of the parameters of the system. We finally validate our analytic results with numerical experiments.

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arXiv2023

Locality and Error Mitigation of Quantum Circuits

In this work, we study and improve two leading error mitigation techniques, namely Probabilistic Error Cancellation (PEC) and Zero-Noise Extrapolation (ZNE), for estimating the expectation value of local observables. For PEC, we introduce a new estimator that takes into account the light cone of the unitary circuit with respect to a target local observable. Given a fixed error tolerance, the sampling overhead for the new estimator can be several orders of magnitude smaller than the standard PEC estimators. For ZNE, we also use light-cone arguments to establish an error bound that closely captures the behavior of the bias that remains after extrapolation.

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arXiv2022

Quantum Non-Demolition Photon Counting in a 2d Rydberg Atom Array

Rydberg arrays merge the collective behavior of ordered atomic arrays with the controllability and optical nonlinearities of Rydberg systems, resulting in a powerful platform for realizing photonic many-body physics. As an application of this platform, we propose a protocol for quantum non-demolition (QND) photon counting. Our protocol involves photon storage in the Rydberg array, an observation phase consisting of a series of Rabi flops to a Rydberg state and measurements, and retrieval of the stored photons. The Rabi frequency experiences a $\sqrt{n}$ collective enhancement, where $n$ is the number of photons stored in the array. Projectively measuring the presence or absence of a Rydberg excitation after oscillating for some time is thus a weak measurement of photon number. We demonstrate that the photon counting protocol can be used to distill Fock states from arbitrary pure or mixed initial states and to perform photonic state discrimination. We confirm that the protocol still works in the presence of experimentally realistic noise.

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arXiv2022

Diagnosing Barren Plateaus with Tools from Quantum Optimal Control

Variational Quantum Algorithms (VQAs) have received considerable attention due to their potential for achieving near-term quantum advantage. However, more work is needed to understand their scalability. One known scaling result for VQAs is barren plateaus, where certain circumstances lead to exponentially vanishing gradients. It is common folklore that problem-inspired ansatzes avoid barren plateaus, but in fact, very little is known about their gradient scaling. In this work we employ tools from quantum optimal control to develop a framework that can diagnose the presence or absence of barren plateaus for problem-inspired ansatzes. Such ansatzes include the Quantum Alternating Operator Ansatz (QAOA), the Hamiltonian Variational Ansatz (HVA), and others. With our framework, we prove that avoiding barren plateaus for these ansatzes is not always guaranteed. Specifically, we show that the gradient scaling of the VQA depends on the degree of controllability of the system, and hence can be diagnosed through the dynamical Lie algebra $\mathfrak{g}$ obtained from the generators of the ansatz. We analyze the existence of barren plateaus in QAOA and HVA ansatzes, and we highlight the role of the input state, as different initial states can lead to the presence or absence of barren plateaus. Taken together, our results provide a framework for trainability-aware ansatz design strategies that do not come at the cost of extra quantum resources. Moreover, we prove no-go results for obtaining ground states with variational ansatzes for controllable system such as spin glasses. Our work establishes a link between the existence of barren plateaus and the scaling of the dimension of $\mathfrak{g}$.

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arXiv2022

Variational Quantum State Eigensolver

Extracting eigenvalues and eigenvectors of exponentially large matrices will be an important application of near-term quantum computers. The Variational Quantum Eigensolver (VQE) treats the case when the matrix is a Hamiltonian. Here, we address the case when the matrix is a density matrix $ρ$. We introduce the Variational Quantum State Eigensolver (VQSE), which is analogous to VQE in that it variationally learns the largest eigenvalues of $ρ$ as well as a gate sequence $V$ that prepares the corresponding eigenvectors. VQSE exploits the connection between diagonalization and majorization to define a cost function $C=\Tr(\tildeρ H)$ where $H$ is a non-degenerate Hamiltonian. Due to Schur-concavity, $C$ is minimized when $\tildeρ = VρV^\dagger$ is diagonal in the eigenbasis of $H$. VQSE only requires a single copy of $ρ$ (only $n$ qubits) per iteration of the VQSE algorithm, making it amenable for near-term implementation. We heuristically demonstrate two applications of VQSE: (1) Principal component analysis, and (2) Error mitigation.

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arXiv2022

Generalization in quantum machine learning from few training data

Modern quantum machine learning (QML) methods involve variationally optimizing a parameterized quantum circuit on a training data set, and subsequently making predictions on a testing data set (i.e., generalizing). In this work, we provide a comprehensive study of generalization performance in QML after training on a limited number $N$ of training data points. We show that the generalization error of a quantum machine learning model with $T$ trainable gates scales at worst as $\sqrt{T/N}$. When only $K \ll T$ gates have undergone substantial change in the optimization process, we prove that the generalization error improves to $\sqrt{K / N}$. Our results imply that the compiling of unitaries into a polynomial number of native gates, a crucial application for the quantum computing industry that typically uses exponential-size training data, can be sped up significantly. We also show that classification of quantum states across a phase transition with a quantum convolutional neural network requires only a very small training data set. Other potential applications include learning quantum error correcting codes or quantum dynamical simulation. Our work injects new hope into the field of QML, as good generalization is guaranteed from few training data.

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arXiv2022

Trainability of Dissipative Perceptron-Based Quantum Neural Networks

Several architectures have been proposed for quantum neural networks (QNNs), with the goal of efficiently performing machine learning tasks on quantum data. Rigorous scaling results are urgently needed for specific QNN constructions to understand which, if any, will be trainable at a large scale. Here, we analyze the gradient scaling (and hence the trainability) for a recently proposed architecture that we called dissipative QNNs (DQNNs), where the input qubits of each layer are discarded at the layer's output. We find that DQNNs can exhibit barren plateaus, i.e., gradients that vanish exponentially in the number of qubits. Moreover, we provide quantitative bounds on the scaling of the gradient for DQNNs under different conditions, such as different cost functions and circuit depths, and show that trainability is not always guaranteed.

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arXiv2022

Optimal tests for continuous-variable quantum teleportation and photodetectors

Quantum teleportation is a primitive in several important applications, including quantum communication, quantum computation, error correction, and quantum networks. In this work, we propose an optimal test for the performance of continuous-variable (CV) quantum teleportation in terms of the energy-constrained channel fidelity between ideal CV teleportation and its experimental implementation. Work prior to ours considered suboptimal tests of the performance of CV teleportation, focusing instead on its performance for particular states, such as ensembles of coherent states, squeezed states, cat states, etc. Here we prove that the optimal state for testing CV teleportation is an entangled superposition of twin Fock states. We establish this result by reducing the problem of estimating the energy-constrained channel fidelity between ideal CV teleportation and its experimental approximation to a quadratic program and solving it. As an additional result, we obtain an analytical solution to the energy-constrained diamond distance between a photodetector and its experimental approximation. These results are relevant for experiments that make use of CV teleportation and photodetectors.

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arXiv2022

Connecting ansatz expressibility to gradient magnitudes and barren plateaus

Parameterized quantum circuits serve as ansätze for solving variational problems and provide a flexible paradigm for programming near-term quantum computers. Ideally, such ansätze should be highly expressive so that a close approximation of the desired solution can be accessed. On the other hand, the ansatz must also have sufficiently large gradients to allow for training. Here, we derive a fundamental relationship between these two essential properties: expressibility and trainability. This is done by extending the well established barren plateau phenomenon, which holds for ansätze that form exact 2-designs, to arbitrary ansätze. Specifically, we calculate the variance in the cost gradient in terms of the expressibility of the ansatz, as measured by its distance from being a 2-design. Our resulting bounds indicate that highly expressive ansätze exhibit flatter cost landscapes and therefore will be harder to train. Furthermore, we provide numerics illustrating the effect of expressiblity on gradient scalings, and we discuss the implications for designing strategies to avoid barren plateaus.

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arXiv2022

Reformulation of the No-Free-Lunch Theorem for Entangled Data Sets

The no-free-lunch (NFL) theorem is a celebrated result in learning theory that limits one's ability to learn a function with a training data set. With the recent rise of quantum machine learning, it is natural to ask whether there is a quantum analog of the NFL theorem, which would restrict a quantum computer's ability to learn a unitary process (the quantum analog of a function) with quantum training data. However, in the quantum setting, the training data can possess entanglement, a strong correlation with no classical analog. In this work, we show that entangled data sets lead to an apparent violation of the (classical) NFL theorem. This motivates a reformulation that accounts for the degree of entanglement in the training set. As our main result, we prove a quantum NFL theorem whereby the fundamental limit on the learnability of a unitary is reduced by entanglement. We employ Rigetti's quantum computer to test both the classical and quantum NFL theorems. Our work establishes that entanglement is a commodity in quantum machine learning.

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arXiv2021

Error mitigation on a near-term quantum photonic device

Photon loss is destructive to the performance of quantum photonic devices and therefore suppressing the effects of photon loss is paramount to photonic quantum technologies. We present two schemes to mitigate the effects of photon loss for a Gaussian Boson Sampling device, in particular, to improve the estimation of the sampling probabilities. Instead of using error correction codes which are expensive in terms of their hardware resource overhead, our schemes require only a small amount of hardware modifications or even no modification. Our loss-suppression techniques rely either on collecting additional measurement data or on classical post-processing once the measurement data is obtained. We show that with a moderate cost of classical post processing, the effects of photon loss can be significantly suppressed for a certain amount of loss. The proposed schemes are thus a key enabler for applications of near-term photonic quantum devices.

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arXiv2020

Information-theoretic aspects of the generalized amplitude damping channel

The generalized amplitude damping channel (GADC) is one of the sources of noise in superconducting-circuit-based quantum computing. It can be viewed as the qubit analogue of the bosonic thermal channel, and it thus can be used to model lossy processes in the presence of background noise for low-temperature systems. In this work, we provide an information-theoretic study of the GADC. We first determine the parameter range for which the GADC is entanglement breaking and the range for which it is anti-degradable. We then establish several upper bounds on its classical, quantum, and private capacities. These bounds are based on data-processing inequalities and the uniform continuity of information-theoretic quantities, as well as other techniques. Our upper bounds on the quantum capacity of the GADC are tighter than the known upper bound reported recently in [Rosati et al., Nat. Commun. 9, 4339 (2018)] for the entire parameter range of the GADC, thus reducing the gap between the lower and upper bounds. We also establish upper bounds on the two-way assisted quantum and private capacities of the GADC. These bounds are based on the squashed entanglement, and they are established by constructing particular squashing channels. We compare these bounds with the max-Rains information bound, the mutual information bound, and another bound based on approximate covariance. For all capacities considered, we find that a large variety of techniques are useful in establishing bounds.

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arXiv2020

Noise Resilience of Variational Quantum Compiling

Variational hybrid quantum-classical algorithms (VHQCAs) are near-term algorithms that leverage classical optimization to minimize a cost function, which is efficiently evaluated on a quantum computer. Recently VHQCAs have been proposed for quantum compiling, where a target unitary $U$ is compiled into a short-depth gate sequence $V$. In this work, we report on a surprising form of noise resilience for these algorithms. Namely, we find one often learns the correct gate sequence $V$ (i.e., the correct variational parameters) despite various sources of incoherent noise acting during the cost-evaluation circuit. Our main results are rigorous theorems stating that the optimal variational parameters are unaffected by a broad class of noise models, such as measurement noise, gate noise, and Pauli channel noise. Furthermore, our numerical implementations on IBM's noisy simulator demonstrate resilience when compiling the quantum Fourier transform, Toffoli gate, and W-state preparation. Hence, variational quantum compiling, due to its robustness, could be practically useful for noisy intermediate-scale quantum devices. Finally, we speculate that this noise resilience may be a general phenomenon that applies to other VHQCAs such as the variational quantum eigensolver.

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arXiv2020

Conditional quantum one-time pad

Suppose that Alice and Bob are located in distant laboratories, which are connected by an ideal quantum channel. Suppose further that they share many copies of a quantum state $ρ_{ABE}$, such that Alice possesses the $A$ systems and Bob the $BE$ systems. In our model, there is an identifiable part of Bob's laboratory that is insecure: a third party named Eve has infiltrated Bob's laboratory and gained control of the $E$ systems. Alice, knowing this, would like use their shared state and the ideal quantum channel to communicate a message in such a way that Bob, who has access to the whole of his laboratory ($BE$ systems), can decode it, while Eve, who has access only to a sector of Bob's laboratory ($E$ systems) and the ideal quantum channel connecting Alice to Bob, cannot learn anything about Alice's transmitted message. We call this task the conditional one-time pad, and in this paper, we prove that the optimal rate of secret communication for this task is equal to the conditional quantum mutual information $I(A;B|E)$ of their shared state. We thus give the conditional quantum mutual information an operational meaning that is different from those given in prior works, via state redistribution, conditional erasure, or state deconstruction. We also generalize the model and method in several ways, one of which demonstrates that the negative tripartite interaction information $-I_{3}(A;B;E) = I(A;BE)-I(A;B)-I(A;E)$ of a tripartite state $ρ_{ABE}$ is an achievable rate for a secret-sharing task, i.e., the case in which Alice's message should be secure from someone possessing only the $AB$ or $AE$ systems but should be decodable by someone possessing all systems $A$, $B$, and $E$.

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arXiv2020

Characterizing the performance of continuous-variable Gaussian quantum gates

The required set of operations for universal continuous-variable quantum computation can be divided into two primary categories: Gaussian and non-Gaussian operations. Furthermore, any Gaussian operation can be decomposed as a sequence of phase-space displacements and symplectic transformations. Although Gaussian operations are ubiquitous in quantum optics, their experimental realizations generally are approximations of the ideal Gaussian unitaries. In this work, we study different performance criteria to analyze how well these experimental approximations simulate the ideal Gaussian unitaries. In particular, we find that none of these experimental approximations converge uniformly to the ideal Gaussian unitaries. However, convergence occurs in the strong sense, or if the discrimination strategy is energy bounded, then the convergence is uniform in the Shirokov-Winter energy-constrained diamond norm and we give explicit bounds in this latter case. We indicate how these energy-constrained bounds can be used for experimental implementations of these Gaussian unitaries in order to achieve any desired accuracy.

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