A Stepwise Planned Approach to the Solution of Hilbert's Sixth Problem. II : Supmech and Quantum Systems

Supmech, which is noncommutative Hamiltonian mechanics \linebreak (NHM) (developed in paper I) with two extra ingredients : positive observable valued measures (PObVMs) [which serve to connect state-induced expectation values and classical probabilities] and the `CC condition' [which stipulates that the sets of observables and pure states be mutually separating] is proposed as a universal mechanics potentially covering all physical phenomena. It facilitates development of an autonomous formalism for quantum mechanics. Quantum systems, defined algebraically as supmech Hamiltonian systems with non-supercommutative system algebras, are shown to inevitably have Hilbert space based realizations (so as to accommodate rigged Hilbert space based Dirac bra-ket formalism), generally admitting commutative superselection rules. Traditional features of quantum mechanics of finite particle systems appear naturally. A treatment of localizability much simpler and more general than the traditional one is given. Treating massive particles as localizable elementary quantum systems, the Schr$\ddot{o}$dinger wave functions with traditional Born interpretation appear as natural objects for the description of their pure states and the Schr$\ddot{o}$dinger equation for them is obtained without ever using a classical Hamiltonian or Lagrangian. A provisional set of axioms for the supmech program is given.Comment: 55 pages; some modifications in text; improved treatment of topological aspects and of Noether invariants; results unchange

Robust quantum spatial search

Quantum spatial search has been widely studied with most of the study focusing on quantum walk algorithms. We show that quantum walk algorithms are extremely sensitive to systematic errors. We present a recursive algorithm which offers significant robustness to certain systematic errors. To search N items, our recursive algorithm can tolerate errors of size O(1/\sqrt{\ln N}) which is exponentially better than quantum walk algorithms for which tolerable error size is only O(\ln N/\sqrt{N}). Also, our algorithm does not need any ancilla qubit. Thus our algorithm is much easier to implement experimentally compared to quantum walk algorithms

Postprocessing can speed up general quantum search algorithms

A general quantum search algorithm aims to evolve a quantum system from a known source state $|s\rangle$ to an unknown target state $|t\rangle$. It uses a diffusion operator $D_{s}$ having source state as one of its eigenstates and $I_{t}$, where $I_{\psi}$ denotes the selective phase inversion of $|\psi\rangle$ state. It evolves $|s\rangle$ to a particular state $|w\rangle$, call it w-state, in $O(B/\alpha)$ time steps where $\alpha$ is $|\langle t|s\rangle|$ and $B$ is a characteristic of the diffusion operator. Measuring the w-state gives the target state with the success probability of $O(1/B^{2})$ and $O(B^{2})$ applications of the algorithm can boost it from $O(1/B^{2})$ to $O(1)$, making the total time complexity $O(B^{3}/\alpha)$. In the special case of Grover's algorithm, $D_{s}$ is $I_{s}$ and $B$ is very close to $1$. A more efficient way to boost the success probability is quantum amplitude amplification provided we can efficiently implement $I_{w}$. Such an efficient implementation is not known so far. In this paper, we present an efficient algorithm to approximate selective phase inversions of the unknown eigenstates of an operator using phase estimation algorithm. This algorithm is used to efficiently approximate $I_{w}$ which reduces the time complexity of general algorithm to $O(B/\alpha)$. Though $O(B/\alpha)$ algorithms are known to exist, our algorithm offers physical implementation advantages.Comment: Accepted for publication in Physical Review A. arXiv admin note: substantial text overlap with arXiv:1210.464

Quantum search algorithm tailored to clause satisfaction problems

Many important computer science problems can be reduced to clause satisfaction problem. We are given $n$ Boolean variables $x_{k}$ and $m$ clauses $c_{j}$ where each clause is a function of values of some of the variables. We want to find an assignment $i$ of variables for which all $m$ clauses are satisfied. Let $f_{j}(i)$ be a binary function which is $1$ if $j^{\rm th}$ clause is satisfied by the assignment $i$ else $f_{j}(i) = 0$. Then the solution is $r$ for which $f(i=r) = 1$, where $f(i)$ is the AND function of all $f_{j}(i)$. In quantum computing, Grover`s algorithm can be used to find $r$. A crucial component of this algorithm is the selective phase inversion $I_{r}$ of the solution state encoding $r$. $I_{r}$ is implemented by computing $f(i)$ for all $i$ in superposition which requires computing AND of all $m$ binary functions $f_{j}(i)$. Hence there must be coupling between the computation circuits for each $f_{j}(i)$. In this paper, we present an alternative quantum search algorithm which relaxes the requirement of such couplings. Hence it offers implementation advantages for clause satisfaction problems