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Indian Institute of Science
1.
Kumar, Parveen.
Quantum dynamics of weak measurements: Understanding the Born rule and applying weak error correction.
Degree: PhD, Faculty of Science, 2019, Indian Institute of Science
URL: http://etd.iisc.ac.in/handle/2005/4261
► Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigen state will…
(more)
▼ Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigen state 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. In this work, I revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. I describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigen states. In this effective theory framework for the ensemble of quantum trajectories, the measurement interaction is specific to each system-apparatus pair—a context necessary for understanding weak measurements. Also in this framework, the constraint to reproduce projective measurement as per the Born rule in the appropriate limit, requires that the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation dissipation relation. This relation implies that both the noise and the attraction have a common origin in the underlying measurement interaction between the system and the apparatus. I 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.
I test the trajectory ensemble distribution predicted by the quantum diffusion model against the experimental data for weak measurement of superconducting transmon qubits. There is a good fit between theory and experiment for different initial states and several weak measurement couplings. This test vindicates the continuous stochastic measurement framework for quantum state collapse, where the rate of collapse is a characteristic parameter for each system-apparatus pair and is not a universal constant. Furthermore, it implies that the environment can influence the measurement outcomes only via the apparatus and not directly. These are important clues in construction of a complete theory of quantum measurement.
The framework of weak measurements can also be used to construct quantum error correction protocols that protect a quantum state from external disturbances. Unlike projective measurements, one can extract only partial information about the error syndrome
from the encoded state using weak measurements. I construct a feedback protocol that probabilistically corrects the error based on the extracted information. Using numerical simulations of one-qubit error correction codes, I show that the 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…
Advisors/Committee Members: Patel, Apoorva D (advisor).
Subjects/Keywords: Quantum mechanics; Quantum computer; Quantum Error correction; Quantum trajectory framework; Research Subject Categories::NATURAL SCIENCES::Physics::Other physics::Computational physics
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APA (6th Edition):
Kumar, P. (2019). Quantum dynamics of weak measurements: Understanding the Born rule and applying weak error correction. (Doctoral Dissertation). Indian Institute of Science. Retrieved from http://etd.iisc.ac.in/handle/2005/4261
Chicago Manual of Style (16th Edition):
Kumar, Parveen. “Quantum dynamics of weak measurements: Understanding the Born rule and applying weak error correction.” 2019. Doctoral Dissertation, Indian Institute of Science. Accessed January 28, 2021.
http://etd.iisc.ac.in/handle/2005/4261.
MLA Handbook (7th Edition):
Kumar, Parveen. “Quantum dynamics of weak measurements: Understanding the Born rule and applying weak error correction.” 2019. Web. 28 Jan 2021.
Vancouver:
Kumar P. Quantum dynamics of weak measurements: Understanding the Born rule and applying weak error correction. [Internet] [Doctoral dissertation]. Indian Institute of Science; 2019. [cited 2021 Jan 28].
Available from: http://etd.iisc.ac.in/handle/2005/4261.
Council of Science Editors:
Kumar P. Quantum dynamics of weak measurements: Understanding the Born rule and applying weak error correction. [Doctoral Dissertation]. Indian Institute of Science; 2019. Available from: http://etd.iisc.ac.in/handle/2005/4261

Indian Institute of Science
2.
Sainadh, U Satya.
An Efficient Quantum Algorithm and Circuit to Generate Eigenstates Of SU(2) and SU(3) Representations.
Degree: MS, Faculty of Science, 2018, Indian Institute of Science
URL: http://etd.iisc.ac.in/handle/2005/3425
► Many quantum computation algorithms, and processes like measurement based quantum computing, require the initial state of the quantum computer to be an eigenstate of a…
(more)
▼ Many quantum computation algorithms, and processes like measurement based quantum computing, require the initial state of the quantum computer to be an eigenstate of a specific unitary operator. Here we study how quantum states that are eigenstates of finite dimensional irreducible representations of the special unitary (SU(d)) and the permutation (S_n) groups can be efficiently constructed in the computational basis formed by tensor products of the qudit states. The procedure is a unitary transform, which first uses Schur-Weyl duality to map every eigenstate to a unique Schur basis state, and then recursively uses the Clebsch - Gordan transform to rotate the Schur basis state to the computational basis. We explicitly provide an efficient quantum algorithm, and the corresponding quantum logic circuit, to generate any desired eigenstate of SU(2) and SU(3) irreducible representations in the computational basis.
Advisors/Committee Members: Patel, Apoorva D (advisor).
Subjects/Keywords: Quantum Mechanics; Quantum Algorithms; Eigenstates - Special Unitary (SU) Group Representations; Quantum Circuits; Special Unitary (SU) Group Representations; Computational Complexity; Schur Transform; Eigenstate - SU(2); Eigenstate - SU(3); High Energy Physics
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
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APA (6th Edition):
Sainadh, U. S. (2018). An Efficient Quantum Algorithm and Circuit to Generate Eigenstates Of SU(2) and SU(3) Representations. (Masters Thesis). Indian Institute of Science. Retrieved from http://etd.iisc.ac.in/handle/2005/3425
Chicago Manual of Style (16th Edition):
Sainadh, U Satya. “An Efficient Quantum Algorithm and Circuit to Generate Eigenstates Of SU(2) and SU(3) Representations.” 2018. Masters Thesis, Indian Institute of Science. Accessed January 28, 2021.
http://etd.iisc.ac.in/handle/2005/3425.
MLA Handbook (7th Edition):
Sainadh, U Satya. “An Efficient Quantum Algorithm and Circuit to Generate Eigenstates Of SU(2) and SU(3) Representations.” 2018. Web. 28 Jan 2021.
Vancouver:
Sainadh US. An Efficient Quantum Algorithm and Circuit to Generate Eigenstates Of SU(2) and SU(3) Representations. [Internet] [Masters thesis]. Indian Institute of Science; 2018. [cited 2021 Jan 28].
Available from: http://etd.iisc.ac.in/handle/2005/3425.
Council of Science Editors:
Sainadh US. An Efficient Quantum Algorithm and Circuit to Generate Eigenstates Of SU(2) and SU(3) Representations. [Masters Thesis]. Indian Institute of Science; 2018. Available from: http://etd.iisc.ac.in/handle/2005/3425

Indian Institute of Science
3.
Rahaman, Md Aminoor.
Search On A Hypercubic Lattice Using Quantum Random Walk.
Degree: MSc Engg, Faculty of Engineering, 2010, Indian Institute of Science
URL: http://etd.iisc.ac.in/handle/2005/972
► Random walks describe diffusion processes, where movement at every time step is restricted only to neighbouring locations. Classical random walks are constructed using the non-relativistic…
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▼ Random walks describe diffusion processes, where movement at every time step is restricted only to neighbouring locations. Classical random walks are constructed using the non-relativistic Laplacian evolution operator and a coin toss instruction. In quantum theory, an alternative is to use the relativistic Dirac operator. That necessarily introduces an internal degree of freedom (chirality), which may be identified with the coin. The resultant walk spreads quadratically faster than the classical one, and can be applied to a variety of graph theoretical problems.
We study in detail the problem of spatial search, i.e. finding a marked site on a hypercubic lattice in d-dimensions. For d=1, the scaling behaviour of classical and quantum spatial search is the same due to the restriction on movement. On the other hand, the restriction on movement hardly matters for d ≥ 3, and scaling behaviour close to Grover’s optimal algorithm(which has no restriction on movement) can be achieved. d=2 is the borderline critical dimension, where infrared divergence in propagation leads to logarithmic slow down that can be minimised using clever chirality flips. In support of these analytic expectations, we present numerical simulation results for d=2 to d=9, using a lattice implementation of the Dirac operator inspired by staggered fermions. We optimise the parameters of the algorithm, and the simulation results demonstrate that the number of binary oracle calls required for d= 2 and d ≥ 3 spatial search problems are O(√NlogN) and O(√N) respectively. Moreover, with increasing d, the results approach the optimal behaviour of Grover’s algorithm(corresponding to mean field theory or d → ∞ limit). In particular, the d = 3 scaling behaviour is only about 25% higher than the optimal value.
Advisors/Committee Members: Patel, Apoorva (advisor).
Subjects/Keywords: Lattice Theory - Data Processing; Quantum Random Walk; Grover's Algorithm; Dirac Operators; Quantum Computation; Dimensional Hypercubic Lattices; Random Walk Algorithm; Spatial Search; Quantum Physics
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Rahaman, M. A. (2010). Search On A Hypercubic Lattice Using Quantum Random Walk. (Masters Thesis). Indian Institute of Science. Retrieved from http://etd.iisc.ac.in/handle/2005/972
Chicago Manual of Style (16th Edition):
Rahaman, Md Aminoor. “Search On A Hypercubic Lattice Using Quantum Random Walk.” 2010. Masters Thesis, Indian Institute of Science. Accessed January 28, 2021.
http://etd.iisc.ac.in/handle/2005/972.
MLA Handbook (7th Edition):
Rahaman, Md Aminoor. “Search On A Hypercubic Lattice Using Quantum Random Walk.” 2010. Web. 28 Jan 2021.
Vancouver:
Rahaman MA. Search On A Hypercubic Lattice Using Quantum Random Walk. [Internet] [Masters thesis]. Indian Institute of Science; 2010. [cited 2021 Jan 28].
Available from: http://etd.iisc.ac.in/handle/2005/972.
Council of Science Editors:
Rahaman MA. Search On A Hypercubic Lattice Using Quantum Random Walk. [Masters Thesis]. Indian Institute of Science; 2010. Available from: http://etd.iisc.ac.in/handle/2005/972

Indian Institute of Science
4.
Tulsi, Tathagat Avatar.
Generalizations Of The Quantum Search Algorithm.
Degree: PhD, Faculty of Science, 2010, Indian Institute of Science
URL: http://etd.iisc.ac.in/handle/2005/951
► Quantum computation has attracted a great deal of attention from the scientific community in recent years. By using the quantum mechanical phenomena of superposition and…
(more)
▼ Quantum computation has attracted a great deal of attention from the scientific community in recent years. By using the quantum mechanical phenomena of superposition and entanglement, a quantum computer can solve certain problems much faster than classical computers. Several quantum algorithms have been developed to demonstrate this quantum speedup. Two important examples are Shor’s algorithm for the factorization problem, and Grover’s algorithm for the search problem. Significant efforts are on to build a large scale quantum computer for implementing these quantum algorithms.
This thesis deals with Grover’s search algorithm, and presents its several generalizations that perform better in specific contexts. While writing the thesis, we have assumed the familiarity of readers with the basics of quantum mechanics and computer
science. For a general introduction to the subject of quantum computation, see [1].
In Chapter 1, we formally define the search problem as well as present Grover’s search algorithm [2]. This algorithm, or more generally the quantum amplitude amplification algorithm [3, 4], drives a quantum system from a prepared initial state (s) to a desired target state (t). It uses O(α-1 = | (t−|s)| -1) iterations of the operator g = IsIt on |s), where { IsIt} are selective phase inversions selective phase inversions of the corresponding states. That is a quadratic speedup over the simple scheme of O(α−2) preparations of |s) and subsequent projective measurements. Several generalizations of Grover’s algorithm exist.
In Chapter 2, we study further generalizations of Grover’s algorithm. We analyse the iteration of the search operator S = DsI t on |s) where Ds is a more general transformation than Is, and I t is a selective phase rotation of |t) by angle . We find sufficient conditions for S to produce a successful quantum search algorithm.
In Chapter 3, we demonstrate that our general framework encapsulates several previous generalizations of Grover’s algorithm. For example, the phase-matching condition for the search operator requires the angles and and to be almost equal for a successful quantum search. In Kato’s algorithm, the search operator is where Ks consists of only single-qubit gates, which are easier to implement physically than multi-qubit gates. The spatial search algorithms consider the search operator where is a spatially local operator and provides implementation advantages over Is. The analysis of Chapter 2 provides a simpler understanding of all these special cases.
In Chapter 4, we present schemes to improve our general quantum search algorithm, by controlling the operators through an ancilla qubit. For the case of two dimensional spatial search problem, these schemes yield an algorithm with time complexity . Earlier algorithms solved this problem in time steps, and it was an open question to design a faster algorithm. The schemes can also be used to find, for a given unitary operator, an eigenstate corresponding to a specified eigenvalue.
In Chapter 5, we extend the analysis…
Advisors/Committee Members: Patel, Apoorva (advisor).
Subjects/Keywords: Quantum Theory; Algorithm; Quantum Search Algorithm; Grover's Search Algorithm; Quantum Computation; Adiabatic Quantum Search; Robust Quantum Search Algorithm; Kato's Algorithm; Fixed-point Quantum Search; Quantum Algorithms; Quantum Mechanics
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Record Details
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Tulsi, T. A. (2010). Generalizations Of The Quantum Search Algorithm. (Doctoral Dissertation). Indian Institute of Science. Retrieved from http://etd.iisc.ac.in/handle/2005/951
Chicago Manual of Style (16th Edition):
Tulsi, Tathagat Avatar. “Generalizations Of The Quantum Search Algorithm.” 2010. Doctoral Dissertation, Indian Institute of Science. Accessed January 28, 2021.
http://etd.iisc.ac.in/handle/2005/951.
MLA Handbook (7th Edition):
Tulsi, Tathagat Avatar. “Generalizations Of The Quantum Search Algorithm.” 2010. Web. 28 Jan 2021.
Vancouver:
Tulsi TA. Generalizations Of The Quantum Search Algorithm. [Internet] [Doctoral dissertation]. Indian Institute of Science; 2010. [cited 2021 Jan 28].
Available from: http://etd.iisc.ac.in/handle/2005/951.
Council of Science Editors:
Tulsi TA. Generalizations Of The Quantum Search Algorithm. [Doctoral Dissertation]. Indian Institute of Science; 2010. Available from: http://etd.iisc.ac.in/handle/2005/951
.