arXiv subjects

Samuel A. Kutin

Publications and source records attributed to Samuel A. Kutin.

6 recordsLinked to original sources

arXiv2026

Permutation Wordle

We introduce a guessing game, permutation Wordle, in which a guesser attempts to recover a hidden permutation in $S_n$. In each round, the guesser guesses a permutation (using information from previous rounds) and is told which entries of that permutation are correct. We describe a natural guessing strategy, which we believe to be optimal. We show that the number of permutations this strategy solves in $k+1$ rounds is the Eulerian number $A(n,k)$. We also describe an extension to suited permutations: the setter chooses a permutation in $S_n$ and also a coloring of $[n]$ using $s$ colors. We generalize our strategy, give a recurrence for the number of suited permutations solved in $k+1$ rounds, and relate these numbers to the Eulerian numbers. In the case of two suits, or signed permutations, we also relate these numbers to the Eulerian numbers of type B.

math.CO↗

arXiv2010

Quantum interpolation of polynomials

We consider quantum interpolation of polynomials. We imagine a quantum computer with black-box access to input/output pairs (x_i, f(x_i)), where f is a degree-d polynomial, and we wish to compute f(0). We give asymptotically tight quantum lower bounds for this problem, even in the case where 0 is among the possible values of x_i.

quant-ph↗

arXiv2007

Computation at a distance

We consider a model of computation motivated by possible limitations on quantum computers. We have a linear array of n wires, and we may perform operations only on pairs of adjacent wires. Our goal is to build a circuits that perform specified operations spanning all n wires. We show that the natural lower bound of n-1 on circuit depth is nearly tight for a variety of problems, and we prove linear upper bounds for additional problems. In particular, using only gates adding a wire (mod 2) into an adjacent wire, we can realize any linear operation in GL_n(2) as a circuit of depth 5n. We show that some linear operations require depth at least 2n+1.

quant-ph↗

arXiv2006

Shor's algorithm on a nearest-neighbor machine

We give a new ``nested adds'' circuit for implementing Shor's algorithm in linear width and quadratic depth on a nearest-neighbor machine. Our circuit combines Draper's transform adder with approximation ideas of Zalka. The transform adder requires small controlled rotations. We also give another version, with slightly larger depth, using only reversible classical gates. We do not know which version will ultimately be cheaper to implement.

quant-ph↗

arXiv2004

A new quantum ripple-carry addition circuit

We present a new linear-depth ripple-carry quantum addition circuit. Previous addition circuits required linearly many ancillary qubits; our new adder uses only a single ancillary qubit. Also, our circuit has lower depth and fewer gates than previous ripple-carry adders.

quant-ph↗

arXiv2004

A logarithmic-depth quantum carry-lookahead adder

We present an efficient addition circuit, borrowing techniques from the classical carry-lookahead arithmetic circuit. Our quantum carry-lookahead (QCLA) adder accepts two n-bit numbers and adds them in O(log n) depth using O(n) ancillary qubits. We present both in-place and out-of-place versions, as well as versions that add modulo 2^n and modulo 2^n - 1. Previously, the linear-depth ripple-carry addition circuit has been the method of choice. Our work reduces the cost of addition dramatically with only a slight increase in the number of required qubits. The QCLA adder can be used within current modular multiplication circuits to reduce substantially the run-time of Shor's algorithm.

quant-ph↗