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Stephen A. Fenner

Publications and source records attributed to Stephen A. Fenner.

13 recordsLinked to original sources
arXiv2026

Parallel Repetition in the Two-Player Quantum Cloning Game

We study parallel repetition in the two-player quantum cloning game, a monogamy-of-entanglement game motivated by quantum position verification. Colisson Palais, Escol\`a-Farr\`as, and Speelman bounded the value of $n$ copies between $(3/4)^n$ and $\cos^{2n}(\pi/8)$. For two copies, we prove that neither bound is tight. An explicit challenge-dependent strategy achieves value $(5+\sqrt{17})/16>9/16$, so strong parallel repetition fails for the unrestricted game. A block Gram matrix argument gives the upper bound $(11+\sqrt{65})/32<\cos^4(\pi/8)$ and strictly improves the previous parallel-repetition upper bound for every $n$. For every $n$, challenge-independent strategies have optimal value $(3/4)^n$.

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arXiv2019

A note on the entangling properties of the C-SIGN and related quantum gates

We show that any essential application of an $n$-qubit C-SIGN or related quantum gate $G$ leaves its qubits everywhere entangled, provided they were not everywhere entangled to begin with. By ``essential'' we mean roughly that $G$ is not equivalent to a similar gate of smaller arity (or to the identity) when applied to the same input state. By ``everywhere entangled'' we mean that the state is not separable with respect to any bipartition of the qubits.

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arXiv2016

Bipartite Perfect Matching is in quasi-NC

We show that the bipartite perfect matching problem is in quasi-NC$^2$. That is, it has uniform circuits of quasi-polynomial size $n^{O(\log n)}$, and $O(log^2 n)$ depth. Previously, only an exponential upper bound was known on the size of such circuits with poly-logarithmic depth. We obtain our result by an almost complete derandomization of the famous Isolation Lemma when applied to yield an efficient randomized parallel algorithm for the bipartite perfect matching problem.

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arXiv2015

Combinatorial Game Complexity: An Introduction with Poset Games

Poset games have been the object of mathematical study for over a century, but little has been written on the computational complexity of determining important properties of these games. In this introduction we develop the fundamentals of combinatorial game theory and focus for the most part on poset games, of which Nim is perhaps the best-known example. We present the complexity results known to date, some discovered very recently.

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arXiv2014

The complexity of some regex crossword problems

In a typical regular expression (regex) crossword puzzle, you are given two nonempty lists $R_1,\ldots,R_m$ and $C_1,\ldots,C_n$ of regular expressions over some alphabet, and your goal is to fill in an $m\times n$ grid with letters from that alphabet so that the string formed by the $i$th row is in $L(R_i)$, and the string formed by the $j$th column is in $L(C_j)$, for all $1\le i\le m$ and $1\le j\le n$. Such a grid is a solution to the puzzle. It is known that determining whether a solution exists is NP-complete. We consider a number of restrictions and variants to this problem where all the $R_i$ are equal to some regular expression $R$, and all the $C_j$ are equal to some regular expression $C$. We call the solution to such a puzzle an $(R,C)$-crossword. Our main results are the following: 1. There exists a fixed regular expression $C$ over the alphabet $\{0,1\}$ such that the following problem is NP-complete: "Given a regular expression $R$ over $\{0,1\}$ and positive integers $m$ and $n$ given in unary, does an $m\times n$ $(R,C)$-crossword exist?" This improves the result mentioned above. 2. The following problem is NP-hard: "Given a regular expression $E$ over $\{0,1\}$ and positive integers $m$ and $n$ given in unary, does an $m\times n$ $(E,E)$-crossword exist?" 3. There exists a fixed regular expression $C$ over $\{0,1\}$ such that the following problem is undecidable (equivalent to the Halting Problem): "Given a regular expression $R$ over $\{0,1\}$, does an $(R,C)$-crossword exist (of any size)?" 4. The following problem is undecidable (equivalent to the Halting Problem): "Given a regular expression $E$ over $\{0,1\}$, does an $(E,E)$-crossword exist (of any size)?"

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arXiv2012

Functions that preserve p-randomness

We show that polynomial-time randomness (p-randomness) is preserved under a variety of familiar operations, including addition and multiplication by a nonzero polynomial-time computable real number. These results follow from a general theorem: If $I$ is an open interval in the reals, $f$ is a function mapping $I$ into the reals, and $r$ in $I$ is p-random, then $f(r)$ is p-random provided 1. $f$ is p-computable on the dyadic rational points in $I$, and 2. $f$ varies sufficiently at $r$, i.e., there exists a real constant $C > 0$ such that either (a) $(f(x) - f(r))/(x-r) > C$ for all $x$ in $I$ with $x \ne r$, or (b) $(f(x) - f(r))(x-r) < -C$ for all $x$ in $I$ with $x \ne r$. Our theorem implies in particular that any analytic function about a p-computable point whose power series has uniformly p-computable coefficients preserves p-randomness in its open interval of absolute convergence. Such functions include all the familiar functions from first-year calculus.

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arXiv2004

Implementing fanout, parity, and Mod gates via spin exchange interactions

We show that, for any n > 0, the Heisenberg interaction among 2n qubits (as spin-1/2 particles) can be used to exactly implement an n-qubit parity gate, which is equivalent in constant depth to an n-qubit fanout gate. Either isotropic or nonisotropic versions of the interaction can be used. We generalize our basic results by showing that any Hamiltonian (acting on suitably encoded logical qubits), whose eigenvalues depend quadratically on the Hamming weight of the logical qubit values, can be used to implement generalized Mod_q gates for any q > 1. This paper is a sequel to quant-ph/0309163, and resolves a question left open in that paper.

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arXiv2003

A note on the classical lower bound for a quantum walk algorithm

A recent paper on quantum walks by Childs et al. [STOC'03] provides an example of a black-box problem for which there is a quantum algorithm with exponential speedup over the best classical randomized algorithm for the problem, but where the quantum algorithm does not involve any use of the quantum Fourier transform. They give an exponential lower bound for a classical randomized algorithm solving the black-box graph traversal problem defined in their paper. In this note we give an improved lower bound for this problem via a straightforward and more complete analysis.

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arXiv2003

Implementing the fanout gate by a Hamiltonian

We show that, for even n, evolving n qubits according to a simple Hamiltonian can be used to exactly implement an (n+1)-qubit parity gate, which is equivalent in constant depth to an (n+1)-qubit fanout gate. We also observe that evolving the Hamiltonian for three qubits results in an inversion-on-three-way-equality gate, which together with single-qubit operations is universal for quantum computation.

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arXiv2003

A Physics-Free Introduction to the Quantum Computation Model

This article defines and proves basic properties of the standard quantum circuit model of computation. The model is developed abstractly in close analogy with (classical) deterministic and probabilistic circuits, without recourse to any physical concepts or principles. It is intended as a primer for theoretical computer scientists who do not know--and perhaps do not care to know--any physics.

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arXiv2001

Universal quantum computation with two- and three-qubit projective measurements

We present a finite set of projective measurements that, together with quantum memory and preparation of the |0> state, suffice for universal quantum computation. This extends work of Nielsen [quant-ph/0108020], who proposed a scheme in which an arbitrary unitary operation on n qubits can be simulated using only projective measurements on at most 2n qubits. All measurements in our set involve two qubits, except two measurements which involve three qubits. Thus we improve by one the upper bound, implied by Nielsen's results, on the maximum number of qubits needed to participate in any single measurement to achieve universal quantum computation. Each of our measurements is two-valued, and each can be expressed mathematically as a Boolean combination of single-qubit measurements.

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arXiv2000

An intuitive Hamiltonian for quantum search

We present new intuition behind Grover's quantum search algorithm by means of a Hamiltonian. Given a black-box Boolean function f mapping strings of length n into {0,1} such that f(w) = 1 for exactly one string w, L. K. Grover describes a quantum algorithm that finds w in O(2^{n/2}) time. Farhi & Gutmann show that w can also be found in the same amount time by letting the quantum system evolve according to a simple Hamiltonian depending only on f. Their system evolves along a path far from that taken by Grover's original algorithm, however. The current paper presents an equally simple Hamiltonian matching Grover's algorithm step for step. The new Hamiltonian is similar in appearance from that of Farhi & Gutmann, but has some important differences, and provides new intuition for Grover's algorithm itself. This intuition both contrasts with and supplements other explanations of Grover's algorithm as a rotation in two dimensions, and suggests that the Hamiltonian-based approach to quantum algorithms can provide a useful heuristic for discovering new quantum algorithms.

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