An algebraic reformulation of the quantum search algorithm associated to a k‐valued oracle function, is introduced in terms of the so called oracle matrix algebra, by means of which a Bloch sphere like description of search is obtained. A parametric family of symmetric completely positive trace preserving (CPTP) maps, that formalize the presence of quantum noise but preserves the complexity of the algorithm, is determined. Dimensional reduction of representations of oracle Lie algebra is introduced in order to determine the reduced density matrix of subsets of qubits in database. The L1 vector‐induced norm of reduced density matrix is employed to define an index function for the quantum entanglement between database qubits, in the presence of non invariant noise CPTP maps. Analytic investigations provide a causal relation between entanglement and fidelity of the algorithm, which is controlled by quantum noise parameter.
An algebraic reformulation of the quantum search algorithm associated to a k‐valued oracle function, is introduced in terms of the so called oracle matrix algebra, by means of which a Bloch sphere like description of search is obtained. A parametric family of symmetric completely positive trace preserving (CPTP) maps, that formalize the presence of quantum noise but preserves the complexity of the algorithm, is determined. Dimensional reduction of representations of oracle Lie algebra is introduced in order to determine the reduced density matrix of subsets of qubits in database. The L1 vector‐induced norm of reduced density matrix is employed to define an index function for the quantum entanglement between database qubits, in the presence of non invariant noise CPTP maps. Analytic investigations provide a causal relation between entanglement and fidelity of the algorithm, which is controlled by quantum noise parameter.
External influences in the form of quantum noise on Grover’s search algorithm are investigated. The study shows that the algorithm can be robust under such external dissipation. The effect of noise is described by a completely positive trace preserving map, acting on an unsorted N‐dimensional database made of projective density matrices. To evaluate the efficiency of the algorithm two figures of merit are used, the radial fidelity giving the projection between the Bloch vectors of target and final density operators, and the cosine of angular fidelity between the same vectors. Explicitly we prove that the resulting search positive map depends on x, the strength of the coupling to the environment, and that there are infinitely many x values, named good x’s and bad x’s, for which search is successful after O(N) queries, or fails, respectively . Next we study the information theoretic aspects of the dissipative quantum search algorithm. To this end we view the initial state as a pure qubit signal entering through a cascade of quantum channels, each channel described by a completely positive map, and corresponds to the iterations of the algorithm. This map is identical to the dissipative search map of the algorithm for which the parameter x now characterizes the influence of the noisy channel exercised upon the incoming quantum signal. The process of searching is understood as transmission of the initial signal/item through the composite channel made by O(N) elementary channels. The quality of this transmission corresponds to the fidelity of targeting the wanted item in the quantum database, and is studied in terms of the majorization/quantum entropy of the input‐output density matrix, and by the entropy exchanged between the qubit signal/item and an external environment.
Employing the single item search algorithm of N dimensional database it is shown that: First, the entanglement developed between two any-size parts of database space varies periodically during the course of searching. The periodic entanglement of the associated reduced density matrix quantified by several entanglement measures (linear entropy, von Neumann, Renyi), is found to vanish with period O(sqrt(N)). Second, functions of equal entanglement are shown to vary also with equal period. Both those phenomena, based on size-independent database bi-partition, manifest a general scale invariant property of entanglement in quantum search. Third, measuring the entanglement periodicity via the number of searching steps between successive canceling out, determines N, the database set cardinality, quadratically faster than ordinary counting. An operational setting that includes an Entropy observable and its quantum circuits realization is also provided for implementing fast counting. Rigging the marked item initial probability, either by initial advice or by guessing, improves hyper-quadratically the performance of those phenomena.
In chapter 2 various parameterizations for the orbits under local unitary transformations of three-qubit pure states are analyzed. It is shown that the entanglement monotones of any multipartite pure state uniquely determine the orbit of that state. It follows that there must be an entanglement monotone for three-qubit pure states which depends on the Kempe invariant defined in [1]. A form for such an entanglement monotone is proposed. A theorem is proved that significantly reduces the number of entanglement monotones that must be looked at to find the maximal probability of transforming one multipartite state
to another. In chapter 3 Grover's unstructured quantum search algorithm is generalized to use an arbitrary starting superposition and an arbitrary unitary matrix. A
formula for the probability of the generalized Grover's algorithm succeeding after n iterations is derived. This formula is used to determine the optimal strategy
for using the unstructured quantum search algorithm. The speedup obtained illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. The analysis
is extended to the case of a society of k quantum searches acting in parallel. In chapter 4 the positive map Г : p → (Trρ) - ρ is introduced as a separability criterion. Any separable state is mapped by the tensor product of Г and the identity in to a non-negative operator, which provides a necessary condition for separability. If Г acts on a two-dimensional subsystem, then it is equivalent to partial transposition and therefore also sufficient for 2 x 2 and 2 x 3 systems.
Finally, a connection between this map for two qubits and complex conjugation in the magic basis [2] is displayed.
We introduce Quantum Index Algebra (QIA) as a finite, index-based algebraic framework for representing and manipulating quantum operators on Hilbert spaces of dimension 2m. In QIA, operators are expressed as structured combinations of basis elements indexed by Boolean codes, allowing products, commutators, and conjugations to be computed through finite rules on discrete indices rather than through dense matrix arithmetic. This representation unifies combinatorial index structure, explicit matrix realization, and transformation properties under Walsh-Hadamard-type transforms within a single formalism. Using QIA and its associated block-matrix realization, we reformulate the Bernstein-Vazirani hidden-string problem in its phase-oracle form entirely within a real, finite-dimensional algebraic setting. We show that, under structured oracle access, the QIA procedure reproduces the Bernstein-Vazirani algorithm exactly and achieves the same asymptotic query complexity and circuit depth as the standard quantum algorithm. In particular, the hidden string is recovered by symbolic manipulation of a sparse algebraic representation of the oracle rather than by numerical simulation of quantum amplitudes. Our results demonstrate that the apparent quantum speed-up in this setting is a consequence of operator structure rather than Hilbert-space dimensionality alone. QIA thus provides a precise language for separating genuinely quantum resources from those arising from algebraic and combinatorial structures and offers a new perspective on the classical simulability of structured quantum circuits.
One specific subset of quantum algorithms is Grovers Ordered Search Problem (OSP), the quantum counterpart of the classical binary search algorithm, which utilizes oracle functions to produce a specified value within an ordered database. Classically, the optimal algorithm is known to have a [Formula: see text] complexity; however, Grovers algorithm has been found to have an optimal complexity between the lower bound of [Formula: see text] and the upper bound of [Formula: see text]. We sought to lower the known upper bound of the OSP. With Farhi et al. MITCTP 2815 (1999), arXiv:quant-ph/9901059], we see that the OSP can be resolved into a translational invariant algorithm to create quantum query algorithm restraints. With these restraints, one can find Laurent polynomials for various [Formula: see text] — queries — and [Formula: see text] — database sizes — thus finding larger recursive sets to solve the OSP and effectively reducing the upper bound. These polynomials are found to be convex functions, allowing one to make use of convex optimization to find an improvement on the known bounds. According to Childs et al. [Phys. Rev. A 75 (2007) 032335], semidefinite programming, a subset of convex optimization, can solve the particular problem represented by the constraints. We were able to implement a program abiding to their formulation of a semidefinite program (SDP), leading us to find that it takes an immense amount of storage and time to compute. To combat this setback, we then formulated an approach to improve results of the SDP using matrix sparsity. Through the development of this approach, along with an implementation of a rudimentary solver, we demonstrate how matrix sparsity reduces the amount of time and storage required to compute the SDP — overall ensuring further improvements will likely be made to reach the theorized lower bound.
We propose an algebraic formulation for two distinct quantum algorithms: a quantum classification algorithm and a quantum search algorithm with a non-uniform initial distribution, both based on Clifford algebras and spinorial representations. In the classification algorithm, we exploit properties of spinorial representations to construct orthogonal quantum states associated with different classes, allowing the identification of an item's class through the evaluation of expectation values of operators derived from the generators of the Clifford algebra. In the quantum search algorithm, we consider a database with prior information in which the oracle is implemented directly using generators of the Clifford algebra, simplifying its realization. The proposed approach provides a unified algebraic description for both algorithms, employing spinorial representations in the construction of quantum states and operators. Computational implementations are presented.