arXiv subjects

Ethan Smith

Publications and source records attributed to Ethan Smith.

At least 37 records · Page 2Linked to original sources

arXiv2021

LEAP: Scaling Numerical Optimization Based Synthesis Using an Incremental Approach

While showing great promise, circuit synthesis techniques that combine numerical optimization with search over circuit structures face scalability challenges due to a large number of parameters, exponential search spaces, and complex objective functions. The LEAP algorithm improves scaling across these dimensions using iterative circuit synthesis, incremental re-optimization, dimensionality reduction, and improved numerical optimization. LEAP draws on the design of the optimal synthesis algorithm QSearch by extending it with an incremental approach to determine constant prefix solutions for a circuit. By narrowing the search space, LEAP improves scalability from four to six qubit circuits. LEAP was evaluated with known quantum circuits such as QFT and physical simulation circuits like the VQE, TFIM, and QITE. LEAP can compile four qubit unitaries up to $59\times$ faster than QSearch and five and six qubit unitaries with up to $1.2\times$ fewer CNOTs compared to the QFAST package. LEAP can reduce the CNOT count by up to $36\times$, or $7\times$ on average, compared to the CQC Tket compiler. Despite its heuristics, LEAP has generated optimal circuits for many test cases with a priori known solutions. The techniques introduced by LEAP are applicable to other numerical-optimization-based synthesis approaches.

quant-ph↗

arXiv2021

Leveraging Randomized Compiling for the QITE Algorithm

The success of the current generation of Noisy Intermediate-Scale Quantum (NISQ) hardware shows that quantum hardware may be able to tackle complex problems even without error correction. One outstanding issue is that of coherent errors arising from the increased complexity of these devices. These errors can accumulate through a circuit, making their impact on algorithms hard to predict and mitigate. Iterative algorithms like Quantum Imaginary Time Evolution are susceptible to these errors. This article presents the combination of both noise tailoring using Randomized Compiling and error mitigation with a purification. We also show that Cycle Benchmarking gives an estimate of the reliability of the purification. We apply this method to the Quantum Imaginary Time Evolution of a Transverse Field Ising Model and report an energy estimation and a ground state infidelity both below 1\%. Our methodology is general and can be used for other algorithms and platforms. We show how combining noise tailoring and error mitigation will push forward the performance of NISQ devices.

quant-ph↗

arXiv2021

Randomized compiling for scalable quantum computing on a noisy superconducting quantum processor

The successful implementation of algorithms on quantum processors relies on the accurate control of quantum bits (qubits) to perform logic gate operations. In this era of noisy intermediate-scale quantum (NISQ) computing, systematic miscalibrations, drift, and crosstalk in the control of qubits can lead to a coherent form of error which has no classical analog. Coherent errors severely limit the performance of quantum algorithms in an unpredictable manner, and mitigating their impact is necessary for realizing reliable quantum computations. Moreover, the average error rates measured by randomized benchmarking and related protocols are not sensitive to the full impact of coherent errors, and therefore do not reliably predict the global performance of quantum algorithms, leaving us unprepared to validate the accuracy of future large-scale quantum computations. Randomized compiling is a protocol designed to overcome these performance limitations by converting coherent errors into stochastic noise, dramatically reducing unpredictable errors in quantum algorithms and enabling accurate predictions of algorithmic performance from error rates measured via cycle benchmarking. In this work, we demonstrate significant performance gains under randomized compiling for the four-qubit quantum Fourier transform algorithm and for random circuits of variable depth on a superconducting quantum processor. Additionally, we accurately predict algorithm performance using experimentally-measured error rates. Our results demonstrate that randomized compiling can be utilized to leverage and predict the capabilities of modern-day noisy quantum processors, paving the way forward for scalable quantum computing.

quant-ph↗

arXiv2021

Constant-Depth Circuits for Dynamic Simulations of Materials on Quantum Computers

Dynamic simulation of materials is a promising application for near-term quantum computers. Current algorithms for Hamiltonian simulation, however, produce circuits that grow in depth with increasing simulation time, limiting feasible simulations to short-time dynamics. Here, we present a method for generating circuits that are constant in depth with increasing simulation time for a subset of one-dimensional materials Hamiltonians, thereby enabling simulations out to arbitrarily long times. Furthermore, by removing the effective limit on the number of feasibly simulatable time-steps, the constant-depth circuits enable Trotter error to be made negligibly small by allowing simulations to be broken into arbitrarily many time-steps. Composed of two-qubit matchgates on nearest-neighbor qubits, these constant-depth circuits are constructed based on a set of multi-matchgate identity relationships. For an $N$-spin system, the constant-depth circuit contains only $\mathcal{O}(N^2)$ CNOT gates. When compared to standard Hamiltonian simulation algorithms, our method generates circuits with order-of-magnitude fewer gates, which allows us to successfully simulate the long-time dynamics of systems with up to 5 spins on available quantum hardware. This paves the way for simulations of long-time dynamics for scientifically and technologically relevant quantum materials, enabling the observation of interesting and important atomic-level physics.

quant-ph↗

arXiv2019

Heuristics for Quantum Compiling with a Continuous Gate Set

We present an algorithm for compiling arbitrary unitaries into a sequence of gates native to a quantum processor. As accurate CNOT gates are hard for the foreseeable Noisy- Intermediate-Scale Quantum devices era, our A* inspired algorithm attempts to minimize their count, while accounting for connectivity. We discuss the search strategy together with metrics to expand the solution frontier. For a workload of circuits with complexity appropriate for the NISQ era, we produce solutions well within the best upper bounds published in literature and match or exceed hand tuned implementations, as well as other existing synthesis alternatives. In particular, when comparing against state-of-the-art available synthesis packages we show 2.4x average (up to 5.3x) reduction in CNOT count. We also show how to re-target the algorithm for a different chip topology and native gate set, while obtaining similar quality results. We believe that empirical tools like ours can facilitate algorithmic exploration, gate set discovery for quantum processor designers, as well as providing useful optimization blocks within the quantum compilation tool-chain.

cs.ET↗

arXiv2016

Sums of Euler products and statistics of elliptic curves

We present several results related to statistics for elliptic curves over a finite field $\mathbb{F}_p$ as corollaries of a general theorem about averages of Euler products that we demonstrate. In this general framework, we can reprove known results such as the average Lang-Trotter conjecture, the average Koblitz conjecture, and the vertical Sato-Tate conjecture, even for very short intervals, not accessible by previous methods. We also compute statistics for new questions, such as the problem of amicable pairs and aliquot cycles, first introduced by Silverman and Stange. Our technique is rather flexible and should be easily applicable to a wide range of similar problems. The starting point of our results is a theorem of Gekeler which gives a reinterpretation of Deuring's theorem in terms of an Euler product involving random matrices, thus making a direct connection between the (conjectural) horizontal distributions and the vertical distributions. Our main technical result then shows that, under certain conditions, a weighted average of Euler products is asymptotic to the Euler product of the average factors.

math.NT↗

arXiv2015

Group structures of elliptic curves over finite fields

It is well-known that if $E$ is an elliptic curve over the finite field $\mathbb{F}_p$, then $E(\mathbb{F}_p)\simeq\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/mk\mathbb{Z}$ for some positive integers $m, k$. Let $S(M,K)$ denote the set of pairs $(m,k)$ with $m\le M$ and $k\le K$ such that there exists an elliptic curve over some prime finite field whose group of points is isomorphic to $\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/mk\mathbb{Z}$. Banks, Pappalardi and Shparlinski recently conjectured that if $K\le (\log M)^{2-ε}$, then a density zero proportion of the groups in question actually arise as the group of points on some elliptic curve over some prime finite field. On the other hand, if $K\ge (\log M)^{2+ε}$, they conjectured that a density one proportion of the groups in question arise as the group of points on some elliptic curve over some prime finite field. We prove that the first part of their conjecture holds in the full range $K\le (\log M)^{2-ε}$, and we prove that the second part of their conjecture holds in the limited range $K\ge M^{4+ε}$. In the wider range $K\ge M^2$, we show that a positive density of the groups in question actually occur.

math.NT↗

arXiv2015

The frequency of elliptic curve groups over prime finite fields

Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field $\mathbb{F}_p$, we study the frequency to which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$. It is well-known that the only permissible groups are of the form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M(G_{m,k})$ be the frequency to which the group $G_{m,k}$ arises in this way. Previously, the second and fourth named authors determined an asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M(G_{m,k})$, pointwise and on average. In particular, we show that $M(G_{m,k})$ is bounded above by a constant multiple of the expected quantity when $m\le k^A$ and that the conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups $G_{m,k}$ when $m\le k^{1/4-ε}$. We also apply our methods to study the frequency to which a given integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.

math.NT↗

arXiv2014

Averages of the number of points on elliptic curves

If $E$ is an elliptic curve defined over $\mathbb Q$ and $p$ is a prime of good reduction for $E$, let $E(\mathbb F_p)$ denote the set of points on the reduced curve modulo $p$. Define an arithmetic function $M_E(N)$ by setting $M_E(N):= \#\{p: \#E(\mathbb F_p)= N\}$. Recently, David and the third author studied the average of $M_E(N)$ over certain "boxes" of elliptic curves $E$. Assuming a plausible conjecture about primes in short intervals, they showed the following: for odd $N$, the average of $M_E(N)$ over a box with sufficiently large sides is $\sim \frac{K^{\ast}(N)}{\log{N}}$ for an explicitly-given function $K^{\ast}(N)$. The function $K^{\ast}(N)$ is somewhat peculiar: defined as a product over the primes dividing $N$, it resembles a multiplicative function at first glance. But further inspection reveals that it is not, and so one cannot directly investigate its properties by the usual tools of multiplicative number theory. In this paper, we overcome these difficulties and prove a number of statistical results about $K^{\ast}(N)$. For example, we determine the mean value of $K^{\ast}(N)$ over all $N$, odd $N$ and prime $N$, and we show that $K^{\ast}(N)$ has a distribution function. We also explain how our results relate to existing theorems and conjectures on the multiplicative properties of $\# E(\mathbb F_p)$, such as Koblitz's conjecture.

math.NT↗

arXiv2014

A Cohen-Lenstra phenomenon for elliptic curves

Given an elliptic curve $E$ and a finite Abelian group $G$, we consider the problem of counting the number of primes $p$ for which the group of points modulo $p$ is isomorphic to $G$. Under a certain conjecture concerning the distribution of primes in short intervals, we obtain an asymptotic formula for this problem on average over a family of elliptic curves.

math.NT↗

arXiv2013

Elliptic curves with a given number of points over finite fields

Given an elliptic curve $E$ and a positive integer $N$, we consider the problem of counting the number of primes $p$ for which the reduction of $E$ modulo $p$ possesses exactly $N$ points over $\mathbb F_p$. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.

math.NT↗

arXiv2013

Average Frobenius distribution for the degree two primes of a number field

Let $K$ be a number field and $r$ an integer. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree two prime ideals of $K$ with trace of Frobenius equal to $r$. Under certain restrictions on $K$, we show that "on average" the number of such prime ideals with norm less than or equal to $x$ satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang-Trotter conjecture and extends the work of several authors.

math.NT↗

arXiv2013

Abelian surfaces over finite fields with prescribed groups

Let A be an abelian surface over F_q, the field of q elements. The rational points on A/\F_q form an abelian group A(\F_q) \simeq \Z/n_1\Z \times \Z/n_1 n_2 \Z \times \Z/n_1 n_2 n_3\Z \times\Z/n_1 n_2 n_3 n_4\Z. We are interested in knowing which groups of this shape actually arise as the group of points on some abelian surface over some finite field. For a fixed prime power q, a characterization of the abelian groups that occur was recently found by Rybakov. One can use this characterization to obtain a set of congruences modulo the integers $n_1, n_2, n_3, n_4$ on certain combinations of coefficients of the corresponding Weil polynomials. We use Rybakov's criterion to show that groups \Z/n_1\Z \times \Z/n_1 n_2 \Z \times \Z/n_1 n_2 n_3\Z \times\Z/n_1 n_2 n_3 n_4\Z do not occur if n_1 is very large with respect to n_2, n_2, n_4 (Theorem \ref{splitbound}), and occur with density zero in a wider range of the variables (Theorem \ref{splitbound-average}).

math.NT↗

arXiv2012

Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals

Let $K$ be a fixed number field, assumed to be Galois over $\mathbb Q$. Let $r$ and $f$ be fixed integers with $f$ positive. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree $f$ prime ideals of $K$ with trace of Frobenius equal to $r$. Except in the case $f=2$, we show that "on average," the number of such prime ideals with norm less than or equal to $x$ satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang-Trotter conjecture and extends the work of several authors.

math.NT↗

arXiv2012

A generalization of the Barban-Davenport-Halberstam Theorem to number fields

For a fixed number field $K$, we consider the mean square error in estimating the number of primes with norm congruent to $a$ modulo $q$ by the Chebotarëv Density Theorem when averaging over all $q\le Q$ and all appropriate $a$. Using a large sieve inequality, we obtain an upper bound similar to the Barban-Davenport-Halberstam Theorem.

math.NT↗

arXiv2012

A Barban-Davenport-Halberstam asymptotic for number fields

Let $K$ be a fixed number field, and assume that $K$ is Galois over $\qq$. Previously, the author showed that when estimating the number of prime ideals with norm congruent to $a$ modulo $q$ via the Chebotarëv Density Theorem, the mean square error in the approximation is small when averaging over all $q\le Q$ and all appropriate $a$. In this article, we replace the upper bound by an asymptotic formula. The result is related to the classical Barban-Davenport-Halberstam Theorem in the case $K=\qq$.

math.NT↗

arXiv2012

Finite field elements of high order arising from modular curves

In this paper, we recursively construct explicit elements of provably high order in finite fields. We do this using the recursive formulas developed by Elkies to describe explicit modular towers. In particular, we give two explicit constructions based on two examples of his formulas and demonstrate that the resulting elements have high order. Between the two constructions, we are able to generate high order elements in every characteristic. Despite the use of the modular recursions of Elkies, our methods are quite elementary and require no knowledge of modular curves. We compare our results to a recent result of Voloch. In order to do this, we state and prove a slightly more refined version of a special case of his result.

math.NT↗

arXiv2012

Elliptic curves, modular forms, and sums of Hurwitz class numbers

Let H(N) denote the Hurwitz class number. It is known that if $p$ is a prime, then {equation*} \sum_{|r|<2\sqrt p}H(4p-r^2) = 2p. {equation*} In this paper, we investigate the behavior of this sum with the additional condition $r\equiv c\pmod m$. Three different methods will be explored for determining the values of such sums. First, we will count isomorphism classes of elliptic curves over finite fields. Second, we will express the sums as coefficients of modular forms. Third, we will manipulate the Eichler-Selberg trace for ula for Hecke operators to obtain Hurwitz class number relations. The cases $m=2,3$ and 4 are treated in full. Partial results, as well as several conjectures, are given for $m=5$ and 7.

math.NT↗