I am a theoretical physicist, currently a Professor at the Centre for HighEnergy Physics, Indian Institute of Science, Bangalore.I have extensively worked on lattice gauge theory investigations ofQuantum Chromodynamics. In recent years, I have been working on algorithms for quantum computers, and application of information theory concepts to understand the structure of genetic languages.
The standard quantum error correction protocols use projective measurements to extract the error ... more 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.
We present a new algorithm for directed quantum search, which is superior to the Phase-π/3 search... more We present a new algorithm for directed quantum search, which is superior to the Phase-π/3 search algorithm [5] in several aspects. In particular, our algorithm does not require complex phases and achieves the optimal reduction in error probability, from ǫ to ǫ 2q+1, for all positive integer values q of oracle queries. (The Phase-π/3 algorithm, requires q = (3 n − 1)/2, with n a positive integer.) In our algorithm, oracle queries and diffusion operations are controlled in a special way by two ancilla qubits, and irreversible measurement operations drive the quantum state towards a fixed point. We analyse various features of the algorithm in detail. 1
Results for light quark masses obtained from lattice QCD simulations are compared and contrasted ... more Results for light quark masses obtained from lattice QCD simulations are compared and contrasted with other determinations. Relevance of these results to estimates of ǫ ′ /ǫ is discussed. 1 Determination of Light Quark Masses Quark masses are not physical observables in QCD, rather they enter the theory as parameters in the Lagrangian. Their values depend on the QCD renormalisation scale, and three quantitative approaches have been used to determine them. 1.1 Chiral perturbation theory This is a low energy (E ≤ 1 GeV) effective field theory of QCD in presence of spontaneous chiral symmetry breaking. With mq ≪ ΛQCD, the pseudo-Nambu-Goldstone bosons are the dominant fields at low energy. O(p2, mq) terms in the effective Lagrangian are fixed by the spectrum. O(p4, p2mq, m2 q) terms are estimated using resonance saturation and large Nc power counting, as well as from phenomenological fits to various form factors. Small electromagnetic and isospin breaking effects are systematically inc...
Isgur-Wise functions parametrise the leading behaviour of weak decay form factors of mesons and b... more Isgur-Wise functions parametrise the leading behaviour of weak decay form factors of mesons and baryons containing a single heavy quark. The form factors for the quark mass operator are calculated in strong coupling lattice QCD, and Isgur-Wise functions extracted from them. Based on renormalisation group invariance of the operators involved, it is argued that the Isgur-Wise functions would be the same in the weak coupling continuum theory. Quantum Chromodynamics with heavy quarks possesses spin-flavour symmetries that become exact as the quark masses go to infinity. These symmetries give rise to relations amongst various matrix elements and form factors of hadrons containing heavy quarks 1. Such relations based on symmetry properties alone are genuine predictions of QCD, and do not suffer from the uncertainties of phenomenological models of hadrons. Of course, the leading order relations (i.e. those valid in the M → ∞ limit) have to be corrected for symmetry breaking effects in orde...
We calculate the perturbative corrections to fermion bilinears that are used in numerical simulat... more We calculate the perturbative corrections to fermion bilinears that are used in numerical simulations when extracting weak matrix elements using staggered fermions. This extends previous calculations of Golterman and Smit, and Daniel and Sheard. In particular, we calculate the corrections for non-local bilinears defined in Landau gauge with gauge links excluded. We do this for the simplest operators, i.e. those defined on a 2 4 hypercube, and for tree level improved operators which live on 4 4 hypercubes. We also consider gauge invariant operators in which the “tadpole ” contributions are suppressed by projecting the sums of products of gauge links back in to the gauge group. In all cases, we find that the variation in the size of the perturbative corrections is smaller than those with the gauge invariant unimproved operators. This is most strikingly true for the smeared operators. We investigate the efficacy of the mean-field method of Lepage and Mackenzie at summing up tadpole con...
I review the current status of several lattice QCD results. I concentrate on new analytical devel... more I review the current status of several lattice QCD results. I concentrate on new analytical developments and on numerical results relevant to phenomenology. 1.
In the last few years, lattice QCD has made a dramatic progress in understanding the physics of h... more In the last few years, lattice QCD has made a dramatic progress in understanding the physics of hadrons containing heavy quarks, from the first principle. This review summarises the major achievements. I. INTRODUCTION Heavy quarks play a dominant role in the understanding of the weak interactions as well as of what may lie beyond the standard model. Since quarks are always bound into hadrons, and we do not understand the strong interactions rigorously, the results of these phenomena are often expressed in terms of non-perturbative parameters (also called matrix elements) that reflect our knowledge/ignorance of the strong interaction effects. Lattice QCD offers the best route to a non-perturbative determination of these parameters. The field of lattice gauge theories has nowadays achieved a maturity level where statistical errors are beaten down enough to expose systematic effects. This control over systematic effects has helped reliable extraction of matrix elements that allow us to...
The standard quantum error correction protocols use projective measurements to extract the error ... more 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.
We have developed a software library that simulates noisy quantum logic circuits. We represent qu... more We have developed a software library that simulates noisy quantum logic circuits. We represent quantum states by their density matrices, and incorporate possible errors in initialisation, logic gates, memory and measurement using simple models. Our quantum simulator is implemented as a new backend on IBM's open-source Qiskit platform. In this document, we provide its description, and illustrate it with some simple examples.
The question of whether quantum spatial search in two dimensions can be made optimal has long bee... more The question of whether quantum spatial search in two dimensions can be made optimal has long been an open problem. We report progress towards its resolution by showing that the oracle complexity for target location can be made optimal, by increasing the number of calls to the walk operator that incorporates the graph structure by a logarithmic factor. Our algorithm does not require amplitude amplification. An important ingredient of our algorithm is the implementation of multi-step quantum walks by graph powering, using a coin space of walk-length dependent dimension, which may be of independent interest. Finally, we demonstrate how to implement quantum walks arising from powers of symmetric Markov chains using our methods.
Execution of Grover's quantum search algorithm needs rather limited resources without much fi... more Execution of Grover's quantum search algorithm needs rather limited resources without much fine tuning. Consequently, the algorithm can be implemented in a wide variety of physical set-ups, which involve wave dynamics but may not need other quantum features. Several of these set-ups are described, pointing out that some of them occur quite naturally. In particular, it is entirely possible that the algorithm played a key role in selection of the universal structure of genetic languages.
The principle of equivalence provides a description of gravity in terms of the metric tensor and ... more The principle of equivalence provides a description of gravity in terms of the metric tensor and determines how gravity affects the light cone structure of the space-time. This, in turn, leads to the existence of observers (in any space-time) who do not have access to regions of space-time bounded by horizons. To take into account this generic possibility, it is necessary to demand that physical theories in a given coordinate system must be formulated entirely in terms of variables that an observer using that coordinate system can access. This principle is powerful enough to obtain the following results: (a) The action principle of gravity must be of such a structure that, in the semiclassical limit, the action of the unobserved degrees of freedom reduces to a boundary contribution $A_{\rm boundary}$ obtained by integrating a four divergence. (b) When the boundary is a horizon, $A_{\rm boundary}$ essentially reduces to a single, well-defined, term. (c) This boundary term must have a...
Results for light quark masses obtained from lattice QCD simulations are compared and contrasted ... more Results for light quark masses obtained from lattice QCD simulations are compared and contrasted with other determinations. Relevance of these results to estimates of epsilon/epsilon is discussed.
Quantum measurements are described as instantaneous projections in textbooks. They can be stretch... more 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 quan...
Quantum searching requires precise knowledge of problem parameters (such as the fraction of targe... more Quantum searching requires precise knowledge of problem parameters (such as the fraction of target states) for efficient operation. Recently an algorithm has been discovered, referred to as the Phase-π/3 search algorithm, which gets around this limitation. This algorithm can search a database with the fraction of target states equal to 1 − ǫ so that in q queries it produces a probability of error equal to ǫ 2q+1 which has since been proved to be optimal. This paper gives a different algorithm which has the same worst-case behavior as the Phaseπ/3 search algorithm but much better average-case behavior. Furthermore the new algorithm gives ǫ 2q+1 convergence for all integral q, the Phase-π/3 search algorithm, requires q to be (3 n − 1)/2, with n a positive integer. In the new algorithm, the operations are controlled in a special way by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations.
We construct a quantum random walk algorithm, based on the Dirac operator instead of the Laplacia... more We construct a quantum random walk algorithm, based on the Dirac operator instead of the Laplacian. The algorithm explores multiple evolutionary branches by superposition of states, and does not require the coin toss instruction of classical randomised algorithms. We use this algorithm to search for a marked vertex on a hypercubic lattice in arbitrary dimensions. Our numerical and analytical results match the scaling behaviour of earlier algorithms that use a coin toss instruction.
The standard quantum error correction protocols use projective measurements to extract the error ... more 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.
We present a new algorithm for directed quantum search, which is superior to the Phase-π/3 search... more We present a new algorithm for directed quantum search, which is superior to the Phase-π/3 search algorithm [5] in several aspects. In particular, our algorithm does not require complex phases and achieves the optimal reduction in error probability, from ǫ to ǫ 2q+1, for all positive integer values q of oracle queries. (The Phase-π/3 algorithm, requires q = (3 n − 1)/2, with n a positive integer.) In our algorithm, oracle queries and diffusion operations are controlled in a special way by two ancilla qubits, and irreversible measurement operations drive the quantum state towards a fixed point. We analyse various features of the algorithm in detail. 1
Results for light quark masses obtained from lattice QCD simulations are compared and contrasted ... more Results for light quark masses obtained from lattice QCD simulations are compared and contrasted with other determinations. Relevance of these results to estimates of ǫ ′ /ǫ is discussed. 1 Determination of Light Quark Masses Quark masses are not physical observables in QCD, rather they enter the theory as parameters in the Lagrangian. Their values depend on the QCD renormalisation scale, and three quantitative approaches have been used to determine them. 1.1 Chiral perturbation theory This is a low energy (E ≤ 1 GeV) effective field theory of QCD in presence of spontaneous chiral symmetry breaking. With mq ≪ ΛQCD, the pseudo-Nambu-Goldstone bosons are the dominant fields at low energy. O(p2, mq) terms in the effective Lagrangian are fixed by the spectrum. O(p4, p2mq, m2 q) terms are estimated using resonance saturation and large Nc power counting, as well as from phenomenological fits to various form factors. Small electromagnetic and isospin breaking effects are systematically inc...
Isgur-Wise functions parametrise the leading behaviour of weak decay form factors of mesons and b... more Isgur-Wise functions parametrise the leading behaviour of weak decay form factors of mesons and baryons containing a single heavy quark. The form factors for the quark mass operator are calculated in strong coupling lattice QCD, and Isgur-Wise functions extracted from them. Based on renormalisation group invariance of the operators involved, it is argued that the Isgur-Wise functions would be the same in the weak coupling continuum theory. Quantum Chromodynamics with heavy quarks possesses spin-flavour symmetries that become exact as the quark masses go to infinity. These symmetries give rise to relations amongst various matrix elements and form factors of hadrons containing heavy quarks 1. Such relations based on symmetry properties alone are genuine predictions of QCD, and do not suffer from the uncertainties of phenomenological models of hadrons. Of course, the leading order relations (i.e. those valid in the M → ∞ limit) have to be corrected for symmetry breaking effects in orde...
We calculate the perturbative corrections to fermion bilinears that are used in numerical simulat... more We calculate the perturbative corrections to fermion bilinears that are used in numerical simulations when extracting weak matrix elements using staggered fermions. This extends previous calculations of Golterman and Smit, and Daniel and Sheard. In particular, we calculate the corrections for non-local bilinears defined in Landau gauge with gauge links excluded. We do this for the simplest operators, i.e. those defined on a 2 4 hypercube, and for tree level improved operators which live on 4 4 hypercubes. We also consider gauge invariant operators in which the “tadpole ” contributions are suppressed by projecting the sums of products of gauge links back in to the gauge group. In all cases, we find that the variation in the size of the perturbative corrections is smaller than those with the gauge invariant unimproved operators. This is most strikingly true for the smeared operators. We investigate the efficacy of the mean-field method of Lepage and Mackenzie at summing up tadpole con...
I review the current status of several lattice QCD results. I concentrate on new analytical devel... more I review the current status of several lattice QCD results. I concentrate on new analytical developments and on numerical results relevant to phenomenology. 1.
In the last few years, lattice QCD has made a dramatic progress in understanding the physics of h... more In the last few years, lattice QCD has made a dramatic progress in understanding the physics of hadrons containing heavy quarks, from the first principle. This review summarises the major achievements. I. INTRODUCTION Heavy quarks play a dominant role in the understanding of the weak interactions as well as of what may lie beyond the standard model. Since quarks are always bound into hadrons, and we do not understand the strong interactions rigorously, the results of these phenomena are often expressed in terms of non-perturbative parameters (also called matrix elements) that reflect our knowledge/ignorance of the strong interaction effects. Lattice QCD offers the best route to a non-perturbative determination of these parameters. The field of lattice gauge theories has nowadays achieved a maturity level where statistical errors are beaten down enough to expose systematic effects. This control over systematic effects has helped reliable extraction of matrix elements that allow us to...
The standard quantum error correction protocols use projective measurements to extract the error ... more 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.
We have developed a software library that simulates noisy quantum logic circuits. We represent qu... more We have developed a software library that simulates noisy quantum logic circuits. We represent quantum states by their density matrices, and incorporate possible errors in initialisation, logic gates, memory and measurement using simple models. Our quantum simulator is implemented as a new backend on IBM's open-source Qiskit platform. In this document, we provide its description, and illustrate it with some simple examples.
The question of whether quantum spatial search in two dimensions can be made optimal has long bee... more The question of whether quantum spatial search in two dimensions can be made optimal has long been an open problem. We report progress towards its resolution by showing that the oracle complexity for target location can be made optimal, by increasing the number of calls to the walk operator that incorporates the graph structure by a logarithmic factor. Our algorithm does not require amplitude amplification. An important ingredient of our algorithm is the implementation of multi-step quantum walks by graph powering, using a coin space of walk-length dependent dimension, which may be of independent interest. Finally, we demonstrate how to implement quantum walks arising from powers of symmetric Markov chains using our methods.
Execution of Grover's quantum search algorithm needs rather limited resources without much fi... more Execution of Grover's quantum search algorithm needs rather limited resources without much fine tuning. Consequently, the algorithm can be implemented in a wide variety of physical set-ups, which involve wave dynamics but may not need other quantum features. Several of these set-ups are described, pointing out that some of them occur quite naturally. In particular, it is entirely possible that the algorithm played a key role in selection of the universal structure of genetic languages.
The principle of equivalence provides a description of gravity in terms of the metric tensor and ... more The principle of equivalence provides a description of gravity in terms of the metric tensor and determines how gravity affects the light cone structure of the space-time. This, in turn, leads to the existence of observers (in any space-time) who do not have access to regions of space-time bounded by horizons. To take into account this generic possibility, it is necessary to demand that physical theories in a given coordinate system must be formulated entirely in terms of variables that an observer using that coordinate system can access. This principle is powerful enough to obtain the following results: (a) The action principle of gravity must be of such a structure that, in the semiclassical limit, the action of the unobserved degrees of freedom reduces to a boundary contribution $A_{\rm boundary}$ obtained by integrating a four divergence. (b) When the boundary is a horizon, $A_{\rm boundary}$ essentially reduces to a single, well-defined, term. (c) This boundary term must have a...
Results for light quark masses obtained from lattice QCD simulations are compared and contrasted ... more Results for light quark masses obtained from lattice QCD simulations are compared and contrasted with other determinations. Relevance of these results to estimates of epsilon/epsilon is discussed.
Quantum measurements are described as instantaneous projections in textbooks. They can be stretch... more 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 quan...
Quantum searching requires precise knowledge of problem parameters (such as the fraction of targe... more Quantum searching requires precise knowledge of problem parameters (such as the fraction of target states) for efficient operation. Recently an algorithm has been discovered, referred to as the Phase-π/3 search algorithm, which gets around this limitation. This algorithm can search a database with the fraction of target states equal to 1 − ǫ so that in q queries it produces a probability of error equal to ǫ 2q+1 which has since been proved to be optimal. This paper gives a different algorithm which has the same worst-case behavior as the Phaseπ/3 search algorithm but much better average-case behavior. Furthermore the new algorithm gives ǫ 2q+1 convergence for all integral q, the Phase-π/3 search algorithm, requires q to be (3 n − 1)/2, with n a positive integer. In the new algorithm, the operations are controlled in a special way by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations.
We construct a quantum random walk algorithm, based on the Dirac operator instead of the Laplacia... more We construct a quantum random walk algorithm, based on the Dirac operator instead of the Laplacian. The algorithm explores multiple evolutionary branches by superposition of states, and does not require the coin toss instruction of classical randomised algorithms. We use this algorithm to search for a marked vertex on a hypercubic lattice in arbitrary dimensions. Our numerical and analytical results match the scaling behaviour of earlier algorithms that use a coin toss instruction.
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