arXiv subjects

Apoorva Patel

Publications and source records attributed to Apoorva Patel.

At least 19 recordsLinked to original sources

arXiv2022

Shot-frugal and Robust quantum kernel classifiers

Quantum kernel methods are a candidate for quantum speed-ups in supervised machine learning. The number of quantum measurements N required for a reasonable kernel estimate is a critical resource, both from complexity considerations and because of the constraints of near-term quantum hardware. We emphasize that for classification tasks, the aim is reliable classification and not precise kernel evaluation, and demonstrate that the former is far more resource efficient. Furthermore, it is shown that the accuracy of classification is not a suitable performance metric in the presence of noise and we motivate a new metric that characterizes the reliability of classification. We then obtain a bound for N which ensures, with high probability, that classification errors over a dataset are bounded by the margin errors of an idealized quantum kernel classifier. Using chance constraint programming and the subgaussian bounds of quantum kernel distributions, we derive several Shot-frugal and Robust (ShofaR) programs starting from the primal formulation of the Support Vector Machine. This significantly reduces the number of quantum measurements needed and is robust to noise by construction. Our strategy is applicable to uncertainty in quantum kernels arising from any source of unbiased noise.

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arXiv2018

Efficient Quantum Algorithms for State Measurement and Linear Algebra Applications

We present an algorithm for measurement of $k$-local operators in a quantum state, which scales logarithmically both in the system size and the output accuracy. The key ingredients of the algorithm are a digital representation of the quantum state, and a decomposition of the measurement operator in a basis of operators with known discrete spectra. We then show how this algorithm can be combined with (a) Hamiltonian evolution to make quantum simulations efficient, (b) the Newton-Raphson method based solution of matrix inverse to efficiently solve linear simultaneous equations, and (c) Chebyshev expansion of matrix exponentials to efficiently evaluate thermal expectation values. The general strategy may be useful in solving many other linear algebra problems efficiently.

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arXiv2018

Quantum Trajectory Distribution for Weak Measurement of a Superconducting Qubit: Experiment meets Theory

Quantum measurements are described as instantaneous projections in textbooks. They can be stretched out in time using weak measurements, whereby one can observe the evolution of a quantum state as it heads towards one of the eigenstates of the measured operator. This evolution can be understood as a continuous nonlinear stochastic process, generating an ensemble of quantum trajectories, consisting of noisy fluctuations on top of geodesics that attract the quantum state towards the measured operator eigenstates. The rate of evolution is specific to each system-apparatus pair, and the Born rule constraint requires the magnitudes of the noise and the attraction to be precisely related. We experimentally observe the entire quantum trajectory distribution for weak measurements of a superconducting qubit in circuit QED architecture, quantify it, and demonstrate that it agrees very well with the predictions of a single-parameter white-noise stochastic process. This characterisation of quantum trajectories is a powerful clue to unraveling the dynamics of quantum measurement, beyond the conventional axiomatic quantum theory.

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arXiv2017

Non-Markovian quantum interference in multilevel quantum systems: Exact master equation approach

We study the non-Markovian quantum interference phenomenon of a multi-state atomic system coupled to a bosonic dissipative environment by using the exact master equations derived in this paper. Two examples involving four-level systems with or without external control fields are investigated. Our results show that non-Markovian master equations are capable of exhibiting quantum interference phenomena in a temporal domain that has not fully explored before. In particular, we show that the environmental memory is instrumental in the onset of non-Markovian quantum interference pattern in different time scales.

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arXiv2017

Weak Measurements, Quantum State Collapse and the Born Rule

Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigenstate will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. We revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. We describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigenstates. Investigation of the restrictions needed on the ensemble of quantum trajectories, so as to reproduce projective measurement in the appropriate limit, shows that the Born rule follows when the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation-dissipation relation. That implies that both the noise and the attraction have a common origin in the measurement interaction between the system and the apparatus. We analyse the quantum trajectory ensemble for the scenarios of quantum diffusion and binary quantum jump, and show that the ensemble distribution is completely determined in terms of a single evolution parameter. This trajectory ensemble distribution can be tested in weak measurement experiments. We also comment on how the required noise may arise in the measuring apparatus.

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arXiv2017

A Complete Characterization of Pretty Good State Transfer on Paths

We give a complete characterization of pretty good state transfer on paths between any pair of vertices with respect to the quantum walk model determined by the XY-Hamiltonian. If $n$ is the length of the path, and the vertices are indexed by the positive integers from 1 to $n$, with adjacent vertices having consecutive indices, then the necessary and sufficient conditions for pretty good state transfer between vertices $a$ and $b$ are that (a) $a + b = n + 1$, (b) $n + 1$ has at most one odd non-trivial divisor, and (c) if $n = 2^t r - 1$, for $r$ odd and $r \neq 1$, then $a$ is a multiple of $2^{t - 1}$.

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arXiv2017

Quantum error correction using weak measurements

The standard quantum error correction protocols use projective measurements to extract the error syndromes from the encoded states. We consider the more general scenario of weak measurements, where only partial information about the error syndrome can be extracted from the encoded state. We construct a feedback protocol that probabilistically corrects the error based on the extracted information. Using numerical simulations of one-qubit error correction codes, we show that our error correction succeeds for a range of the weak measurement strength, where (a) the error rate is below the threshold beyond which multiple errors dominate, and (b) the error rate is less than the rate at which weak measurement extracts information. It is also obvious that error correction with too small a measurement strength should be avoided.

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arXiv2017

Optimisation of Quantum Hamiltonian Evolution: From Two Projection Operators to Local Hamiltonians

Given a quantum Hamiltonian and its evolution time, the corresponding unitary evolution operator can be constructed in many different ways, corresponding to different trajectories between the desired end-points and different series expansions. A choice among these possibilities can then be made to obtain the best computational complexity and control over errors. It is shown how a construction based on Grover's algorithm scales linearly in time and logarithmically in the error bound, and is exponentially superior in error complexity to the scheme based on the straightforward application of the Lie-Trotter formula. The strategy is then extended first to simulation of any Hamiltonian that is a linear combination of two projection operators, and then to any local efficiently computable Hamiltonian. The key feature is to construct an evolution in terms of the largest possible steps instead of taking small time steps. Reflection operations and Chebyshev expansions are used to efficiently control the total error on the overall evolution, without worrying about discretisation errors for individual steps. We also use a digital implementation of quantum states that makes linear algebra operations rather simple to perform.

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arXiv2015

An Evolutionary Formalism for Weak Quantum Measurements

Unitary evolution and projective measurement are fundamental axioms of quantum mechanics. Even though projective measurement yields one of the eigenstates of the measured operator as the outcome, there is no theory that predicts which eigenstate will be observed in which experimental run. There exists only an ensemble description, which predicts probabilities of various outcomes over many experimental runs. We propose a dynamical evolution equation for the projective collapse of the quantum state in individual experimental runs, which is consistent with the well-established framework of quantum mechanics. In case of gradual weak measurements, its predictions for ensemble evolution are different from those of the Born rule. It is an open question whether or not suitably designed experiments can observe this alternate evolution.

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arXiv2015

Optimisation of Quantum Evolution Algorithms

Given a quantum Hamiltonian and its evolution time, the corresponding unitary evolution operator can be constructed in many different ways, corresponding to different trajectories between the desired end-points. A choice among these trajectories can then be made to obtain the best computational complexity and control over errors. As an explicit example, Grover's quantum search algorithm is described as a Hamiltonian evolution problem. It is shown that the computational complexity has a power-law dependence on error when a straightforward Lie-Trotter discretisation formula is used, and it becomes logarithmic in error when reflection operators are used. The exponential change in error control is striking, and can be used to improve many importance sampling methods. The key concept is to make the evolution steps as large as possible while obeying the constraints of the problem. In particular, we can understand why overrelaxation algorithms are superior to small step size algorithms.

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arXiv2012

Baryon Number Correlations in Heavy Ion Collisions

The cross-over region of the quark-gluon plasma (QGP) created in heavy ion collisions is influenced by the nearby deconfinement, chiral and baryon condensation phase transitions. A characteristic signature of the deconfinement transition in this region can be infered using the flux tube model, which is dual to the Polyakov loop description and which offers a visual picture of what happens during the transition. The three-point (anti)vertices of a flux tube network lead to formation of (anti)baryons upon hadronisation. Since there is no fundamental interaction associated with the baryon number, correlations in the baryon number distribution at the last scattering surface directly reflect the preceding pattern of the flux tube vertices in the QGP. An alternating pattern of vertices and antivertices would lead to an oscillatory signal in the two-point baryon number correlations, under the experimental conditions prevalent in heavy ion collisions at RHIC and LHC. The strength of the oscillations is a measure of the flexibility of the QGP.

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arXiv2012

Search on a Fractal Lattice using a Quantum Random Walk

The spatial search problem on regular lattice structures in integer number of dimensions $d\geq2$ has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here we investigate the spatial search problem on fractals of non-integer dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (and not the fractal) dimension. Our numerical results have been obtained on the well-known Sierpinski gaskets in two and three dimensions.

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arXiv2012

Evolution of Quantum Discord and its Stability in Two-Qubit NMR Systems

We investigate evolution of quantum correlations in ensembles of two-qubit nuclear spin systems via nuclear magnetic resonance techniques. We use discord as a measure of quantum correlations and the Werner state as an explicit example. We first introduce different ways of measuring discord and geometric discord in two-qubit systems and then describe the following experimental studies: (a) We quantitatively measure discord for Werner-like states prepared using an entangling pulse sequence. An initial thermal state with zero discord is gradually and periodically transformed into a mixed state with maximum discord. The experimental and simulated behavior of rise and fall of discord agree fairly well. (b) We examine the efficiency of dynamical decoupling sequences in preserving quantum correlations. In our experimental setup, the dynamical decoupling sequences preserved the traceless parts of the density matrices at high fidelity. But they could not maintain the purity of the quantum states and so were unable to keep the discord from decaying. (c) We observe the evolution of discord for a singlet-triplet mixed state during a radio-frequency spin-lock. A simple relaxation model describes the evolution of discord, and the accompanying evolution of fidelity of the long-lived singlet state, reasonably well.

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arXiv2012

Flux Tube Model Signals for Baryon Correlations in Heavy Ion Collisions

The flux tube model offers a pictorial description of what happens during the deconfinement phase transition in QCD. The 3-point vertices of a flux tube network lead to formation of baryons upon hadronisation. Therefore, correlations in the baryon number distribution at the last scattering surface are related to the preceding pattern of the flux tube vertices, and provide a signature of the nearby deconfinement phase transition. I discuss the nature of the expected signal, which should be observable in heavy ion collisions at RHIC and LHC.

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arXiv2011

Quantum Computation: Particle and Wave Aspects of Algorithms

The driving force in the pursuit for quantum computation is the exciting possibility that quantum algorithms can be more efficient than their classical analogues. Research on the subject has unraveled several aspects of how that can happen. Clever quantum algorithms have been discovered in recent years, although not systematically, and the field remains under active investigation. Richard Feynman was one of the pioneers who foresaw the power of quantum computers. In this issue dedicated to him, I give an introduction to how particle and wave aspects contribute to the power of quantum computers. Shor's and Grover's algorithms are analysed as examples.

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arXiv2011

Quantum Algorithms: Database Search and its Variations

The driving force in the pursuit for quantum computation is the exciting possibility that quantum algorithms can be more efficient than their classical analogues. Research on the subject has unraveled several aspects of how that can happen. Clever quantum algorithms have been discovered in recent years, although not systematically, and the field remains under active investigation. This article is an introduction to the quantum database search algorithm. Its extension to the quantum spatial search algorithm is also described.

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arXiv2010

Search on a Hypercubic Lattice using a Quantum Random Walk: I. d>2

Random walks describe diffusion processes, where movement at every time step is restricted to only the neighbouring locations. We construct a quantum random walk algorithm, based on discretisation of the Dirac evolution operator inspired by staggered lattice fermions. We use it to investigate the spatial search problem, i.e. finding a marked vertex on a $d$-dimensional hypercubic lattice. The restriction on movement hardly matters for $d>2$, and scaling behaviour close to Grover's optimal algorithm (which has no restriction on movement) can be achieved. Using numerical simulations, we optimise the proportionality constants of the scaling behaviour, and demonstrate the approach to that for Grover's algorithm (equivalent to the mean field theory or the $d\to\infty$ limit). In particular, the scaling behaviour for $d=3$ is only about 25% higher than the optimal $d\to\infty$ value.

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arXiv2010

Search on a Hypercubic Lattice through a Quantum Random Walk: II. d=2

We investigate the spatial search problem on the two-dimensional square lattice, using the Dirac evolution operator discretised according to the staggered lattice fermion formalism. $d=2$ is the critical dimension for the spatial search problem, where infrared divergence of the evolution operator leads to logarithmic factors in the scaling behaviour. As a result, the construction used in our accompanying article \cite{dgt2search} provides an $O(\sqrt{N}\log N)$ algorithm, which is not optimal. The scaling behaviour can be improved to $O(\sqrt{N\log N})$ by cleverly controlling the massless Dirac evolution operator by an ancilla qubit, as proposed by Tulsi \cite{tulsi}. We reinterpret the ancilla control as introduction of an effective mass at the marked vertex, and optimise the proportionality constants of the scaling behaviour of the algorithm by numerically tuning the parameters.

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