Quantum algorithms for condensed-matter physics, number theory and quantum machine learning

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Quantum Algorithms for Condensed-Matter Physics, Number Theory and Quantum

Machine Learning

Diego Garc´ıa Mart´ın

Supervised by:

Germ´ an Sierra Rodero Jos´ e Ignacio Latorre Sent´ıs

Ph.D. thesis

September, 2022

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Quantum Algorithms for Condensed-Matter Physics, Number Theory and Quantum

Machine Learning

Diego Garc´ıa Mart´ın

Programa de Doctorado en F´ısica Te´ orica Departamento de F´ısica Te´ orica

Facultad de Ciencias

Directores: Germ´ an Sierra Rodero

Jos´ e Ignacio Latorre Sent´ıs

Tutora: Francesca Maria Marchetti

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4

Acknowledgments

Esta tesis doctoral no habr´ıa sido posible sin el apoyo y la ayuda de mucha gente, a la que me gustar´ıa darle las gracias.

En primer lugar quiero agradecerle a mis directores Germán Sierra y José Ignacio Latorre la maravillosa oportunidad que me brindaron al introducirme en el ámbito profesional de la Ciencia. Poder dedicarme a algo que me apasiona y me interesa profundamente es un lujo que es dif´ıcil sobrevalorar. Les estaré eternamente agradecido por ello. Gracias Germán por proponerme empezar el doctorado contigo después de aquel trabajo de fin de máster en el que jugamos por primera vez con el primer ordenador cuántico de acceso libre de la Historia.

Gracias por tu infinita generosidad al publicitarme ante José Ignacio para que me acogiera si surg´ıa la posibilidad de financiar mi doctorado en Barcelona. Fi- nalmente surgió dicha oportunidad, y tú altruistamente me dejaste marchar para que continuara mi camino. Afortunadamente, mi camino no se desligó comple- tamente del tuyo, y tiempo después volvimos a colaborar. Gracias también por introducirme al maravilloso mundo de la Teor´ıa de Números. Gracias José Igna- cio, por todo. Por acogerme como tu doctorando en Barcelona, por enseñarme y compartir tu sabidur´ıa conmigo, por la forma tan inspiradora que tienes de transmitir el significado de una profesión tan especial como la muestra. Por tu concepción de la Ciencia como una labor colectiva de la Humanidad, por encima de los egos o las rivalidades geopol´ıticas. Gracias por Benasque. Gracias por Abu Dhabi, gracias por Singapur.

Me gustar´ıa dar las gracias tambi´en de manera especial a Artur Garcia-Saez.

Tu generosidad posibilitó que yo siguiera con el doctorado en Barcelona una vez se acabó la financiación de los primeros dos años. A pesar de que no colaboramos a nivel cient´ıfico de manera directa, estoy seguro de que habrá ocasión de hacerlo en el futuro, y estoy deseando que as´ı sea. Gracias asimismo a Jorge Cortada y a Jordi Planagumà por los dos primeros años del doctorado en Barcelona.

No pod´ıa faltar tampoco un sentido agradecimiento a mis compañeros y amigos Carlos Bravo-Prieto, Adrián Pérez-Salinas, Sergi Ramos-Calderer, Alba Cervera-Lierta, David López-Núñez y Fabian Zwiehoff. De todos vosotros he aprendido, y con todos he disfrutado de muchos momentos que guardo con cariño en la memoria.

Gracias por supuesto también a todos los compañeros de profesión que se han cruzado en mi camino durante estos años y me han enseñado tantas cosas in- teresantes. Gracias Stefano Carrazza, Mart´ın Larocca, Leandro Aolita. Thank you Ingo Roth for those long and insightful conversations. Thank you to all the family at TII. It was a great experience for me getting to know you all. Thank you Lukasz Cincio and Patrick Coles for giving me the opportunity of partic- ipating in the Quantum Computing Summer School at Los Alamos National Laboratory. It’s been amazing.

Muchas gracias tambi´en a Marco Cerezo. Gracias por todas las ense˜nanzas cient´ıficas, y gracias sobre todo por la confianza depositada al apostar por m´ı

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5 para realizar el postdoc en Los Alamos. Es una magn´ıfica oportunidad que te agradezco enormemente. Tenemos mucha Ciencia por delante.

Finalmente, quiero dar las gracias a todas aquellas personas fuera del ámbito de la investigación que dan sentido a mi vida y me han permitido ser quien soy hoy. Much´ısimas gracias a las personas más importantes de mi vida, mis padres, Luis Alberto y Mar´ıa Jesús, por dármelo y enseñármelo todo. Sin ellos no estar´ıa aqu´ı. Gracias a mi hermano, Raúl, por el cariño y por el afecto, y por toda una vida juntos. Gracias a todos mis amigos y amigas, por ser esa parte tan fundamental de mi vida sin la que casi nada tendr´ıa sentido. Ellos y ellas saben perfectamente quiénes son.

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List of publications

Peer-reviewed

• Carlos Bravo-Prieto, Diego Garc´ıa-Mart´ın and Jos´e I. Latorre. “Quantum Singular Value Decomposer”. Physical Review A 101.6 (2020), p. 062310.

• Diego Garc´ıa-Mart´ın, Eduard Ribas, Stefano Carrazza, Jos´e I. Latorre and Germ´an Sierra. “The Prime state and its quantum relatives”. Quantum 4 (2020), p. 471.

• Stavros Efthymiou, Sergi Ramos-Calderer, Carlos Bravo-Prieto, Adrián Pérez-Salinas, Diego Garc´ıa-Mart´ın, Artur Garcia-Saez, José I. Latorre and Stefano Carrazza. “Qibo: a framework for quantum simulation with hardware acceleration”. Quantum Science and Technology 7.1 (2021), p.

015018.

• Alejandro Sopena, Max Hunter-Gordon, Diego Garc´ıa-Mart´ın, Germ´an Sierra and Esperanza L´opez. “Algebraic Bethe Circuits”. Quantum 6 (2022), p. 796.

ArXiv preprints

• Mart´ın Larocca, Nathan Ju, Diego Garc´ıa-Mart´ın, Patrick J. Coles and Marco Cerezo. “Theory of overparametrization in quantum neural networks”. arXiv preprint arXiv:2109.11676.

• Diego Garc´ıa-Mart´ın, Matthias C. Caro, Marco Cerezo, Lukasz Cincio and Patrick J. Coles. “Likelihood scaling and optimal evolution time in Bayesian Hamiltonian learning”. To appear very soon on the arXiv.

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Open-source software

• Stefano Carrazza and Diego Garc´ıa-Mart´ın. qiboteam/qprime: qprime 1.0.0. Version v1.0.0. 2020.

• Stavros Efthymiou, Stefano Carrazza, Andrea Pasquale, Carlos Bravo- Prieto, AdrianPerezSalinas, Sergi Ramos, Diego Garc´ıa- Mart´ın, Marco Lazzarin, shangtai, Nicole Zattarin, Paul, Adrian Mak, Javier Serrano, and atomicprinter. qiboteam/qibo: Qibo 0.1.8 Version v0.1.8. 2022.

• Diego Garc´ıa-Mart´ın, DiegoGM91/Theory-of-overparametrization. 2022.

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Abstract

Quantum computing is a novel paradigm that seeks to harness quantum mechanical effects like superposition, entanglement and interference to outperform classical computers based upon bits (i.e. zeroes and ones) on certain tasks.

Research into the field of quantum algorithms has produced some exponential speed-ups over the best classical algorithms currently known, with potential applications far beyond the academic domain. Yet, the quest for quantum algorithms that dramatically outperform their classical counterparts has proved to be hard. New quantum algorithms are needed, together with a better under- standing of the type of problems quantum computers excel at.

In this thesis, we explore the capabilities that quantum computers may offer to Condensed-Matter Physics, Number Theory and Quantum Machine Learn- ing. We do so by introducing novel quantum algorithms, both exact and variational, in Chapters 2, 3 and 5. Moreover, we provide in Chapter 4 the first theoretical study of the so-called overparametrization phenomenon in variational quantum circuits, which is likely to play a relevant role in Quantum Machine Learning. In Chapter6, we analyze the scalability of certain Hamiltonian learning quantum algorithms that rely on Bayesian inference to a large number of qubits.

Chapter2also contains original contributions to Quantum Number Theory, a novel approach to Number Theory that employs the tools provided by the formalism of Quantum Information Theory to study mathematical objects like the prime numbers. Finally, in AppendixAwe introduce qibo, an open-source, high-performance library for the classical simulation of quantum circuits, that has been used in most of the works presented in this thesis.

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Resumen

La computación cuántica supone un nuevo paradigma que busca aprovechar efectos cuánticos tales como la superposición, el entrelazamiento y la interfer- encia para superar en ciertas tareas a los ordenadores clásicos, basados en bits (ceros y unos). La investigación en el campo de los algoritmos cuánticos ha dado como resultado ventajas exponenciales respecto de los mejores algoritmos clásicos conocidos hoy en d´ıa, con aplicaciones que van mucho más allá del do- minio académico. Sin embargo, la búsqueda de nuevos algoritmos cuánticos que superen sobradamente a los algoritmos clásicos ha resultado ser complicada. Es necesario encontrar más algoritmos cuánticos, as´ı como conseguir un mejor en- tendimiento de cuáles son los problemas en los que los ordenadores cuánticos sobresalen.

En esta tesis doctoral, se exploran las capacidades que los ordenadores cuánticos pueden ofrecer en problemas de F´ısica de la Materia Condensada, Teor´ıa de Números y Aprendizaje de Máquina Cuántica. Para ello, se introducen algoritmos novedosos, tanto exactos como variacionales, en los Cap´ıtulos 2, 3 y5. Además, se lleva a cabo el primer estudio teórico del llamado fenómeno de sobre-parametrización en circuitos cuánticos variacionales, que con toda proba- bilidad será un fenómeno muy relevante en el Aprendizaje de Máquina Cuántica.

En el Cap´ıtulo 6, se analiza la escalabilidad a un número elevado de cúbits de ciertos algoritmos cuánticos basados en inferencia Bayesiana y empleados para aprender Hamiltonianos cuánticos.

Asimismo, el Cap´ıtulo 2 contiene contribuciones originales al campo de la Teor´ıa de Números Cuántica, un enfoque novedoso a la Teor´ıa de Números que emplea las herramientas proporcionadas por la Teor´ıa de la Información Cuántica para estudiar objetos matemáticos como los números primos. Final- mente, en el ApéndiceAse introduce qibo, una librer´ıa de código abierto y alto rendimiento para la simulación de circuitos cuánticos, la cual ha sido utilizada repetidamente en el resto de trabajos incluidos en esta tesis.

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Introduction

“If the business of physics is ever finished, the world will be a much less interesting place in which to live”

John Gribbin

One of the most fundamental questions in Nature is: given the laws of Physics in our Universe, what are the limits that they impose on what can be known, or more concretely, computed? This is a very deep question, and a complete answer to it is not available.

This question was at the the heart of the origins of quantum computing, which date back to the early 80s. It is often reckoned how the great physi- cist Richard Feynman gave a talk in 1981 at Endicott House (Massachusetts Institute of Technology, or MIT for short), where he explained the limitations of classical computers to simulate Quantum Mechanics due to the exponential growth in memory incurred by the classical description of a quantum many- body system [Fey82]. This limitation had also been pointed out by the Soviet mathematician Yuri Manin in 1980 [Man80]. Feynman went on to suggest that a new type of computer, made out of components that obey the laws of Quantum Mechanics, would be able to efficiently emulate another quantum-mechanical system, as they are both governed by the same rules. Later on, he expanded on his proposal of how to build such a quantum computer [Fey86], extending the ideas proposed by Paul Benioff [Ben80; Ben82]. This landmark is usually considered the beginning of the field of quantum computation.

The next big step forward was provided by David Deutsch in 1985 [Deu85], when he introduced the concept of a quantum Turing machine. He also showed for the first time that a quantum computer could do something faster than any classical computer in a black-box scenario, namely, determine whether a boolean function on a single bit is constant or balanced. The algorithm was later gen- eralized in 1992 by Deutsch himself and Richard Jozsa to boolean functions acting on any number of bits [DJ92]. The Deutsch-Jozsa algorithm provides

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16 CHAPTER 1. INTRODUCTION an oracle separation between EQP, the class of decision problems that can be solved exactly in polynomial time on a quantum computer, and P, the class of decision problems that can be solved exactly in polynomial time on a deterministic classical computer. However, the problem is easy to solve making use of randomness on a classical computer, i.e. the Deutsch-Jozsa algorithm does not provide a separation between EQP and BPP, the class of decision problems that can be solved with an error probability bounded by ¹₃ in polynomial time on a probabilistic classical computer. The first separation between quantum and classical machines was provided by the Bernstein-Vazirani algorithm in 1992 [BV93]. This algorithm proves an oracle separation between BQP, the class of decision problems that can be solved with an error probability bounded by ¹₃ in polynomial time on a quantum computer, and BPP. However, this separation is only polynomial. It was Daniel Simon, in 1994, who proved that an exponential oracle separation between BQP and BPP exists [Sim94].

Simon used the quantum Fourier transform to find the period of function over a binary vector space. Inspired by Simon’s algorithm [Sho22], in 1994 Peter Shor came up with quantum algorithms to compute discrete logarithms and factorize integers exponentially faster on a quantum computer than the best classical algorithms currently known [Sho94]. This was a real breakthrough, because it implied that a functional quantum computer could break many of the existing public-key cryptosystems all over the world. This is still true nowadays.

However, a great barrier was standing before the dream of quantum computing could become a reality, if so in theory. It was clear that errors would occur on the necessarily imperfect quantum devices that we might be able to build, so that they could accumulate and destroy the computations. Within classical information theory, error-correcting codes had been developed for classical devices. The basic principle behind these codes is to encode information redun- dantly, such that it could be recovered at a later time even if it was corrupted by errors. The prospects for achieving error correction for quantum computers were daunting, because of three reasons. First, a quantum system cannot be observed (to check if an error has occurred during a computation) without destroying its state prior to the measurement. Second, errors in quantum computers are con- tinuous, as opposed to the discretized errors that occur in classical bits. And third, there exists a result due to Wootters and Zurek from 1982 [WZ82], called the no-cloning theorem, which says that it is not possible to copy arbitrary quantum states. This prevents the straightforward redundancy strategy based on copying information followed by classical error-correcting repetition codes.

Nevertheless, in an impressive series of discoveries made in consecutive years, Peter Shor showed in 1995 that quantum error-correcting codes are possible [Sho95], and in 1996 that quantum computing gates can be made fault-tolerant [Sho96]. This implies that unwanted errors can be corrected using codes that employ imperfect quantum gates. Building on Shor’s discovery, a threshold theorem was soon proved [ABO97]. This theorem states that, if the error rate in individual quantum gates is below a certain constant threshold, any (arbi- trarily long) quantum circuit can be made fault tolerant with no more than a polynomial extra overhead. Fault-tolerant quantum computation was therefore

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17 feasible, at least in principle. As John Preskill put it [Pre98]: “it is possible to fight entanglement with entanglement”.

These theoretical findings boosted the interest on quantum computers, and governments and companies alike started funding research heavily. Great progress has been made since the times of the first big discoveries by the “founding fathers”, mainly experimentally but also in theory. The first prototypes of quantum computers have been built, using different physical platforms such as superconducting circuits [Bra+22], trapped ions [Bru+19], photons [Mad+22], semiconductors [Zwa+13], etc. Some of them have been made available to re- searchers and also the general public on the cloud. Recently, quantum advantage has been claimed by the Google team [Aru+19] and others [Mad+22]. Quantum advantage refers to the possibility of performing a computation that would be intractable even for the largest supercomputers in the world. Recent advances however in the classical simulation of quantum circuits using tensor networks make the advantage frontier a moving target [Hua+21; Liu+21]. In any case, these experiments showcase the spectacular development that quantum tech- nologies have experienced in the last years.

Yet, these experiments have no known practical application, being just the generation of samples from a probability distribution which is classically hard to sample from. Hence, one of the most pressing questions nowadays is whether a practically-useful quantum advantage can be achieved in the near term, before quantum error correction becomes available. This stage prior to fault-tolerant computation has been dubbed the Noisy Intermediate-Scale Quantum era by Preskill [Pre18], because current devices (and those in the near future) suffer from significant noise and consist of only a few tens or hundreds of qubits.

It is fair to say that it is unclear at the moment whether such quantum advantage can be achieved or not with NISQ devices, but a lot of research effort is going into trying to answer this question. A promising type of algorithms for the NISQ era are the so-called Variational Quantum Algorithms (VQAs) [Cer+21c;

Bha+22], which are inspired by the astounding success of Classical Machine Learning, specially neural networks [Abi+18]. These are hybrid algorithms that employ a parametrized quantum circuit to solve a task of interest; the optimal parameters are learned in a variational fashion by minimizing a given cost function using a classical computer.

In this scenario is where this Ph.D. thesis is framed. Here, we explore the possibilities that quantum computers may provide in a varied set of contexts, ranging from Number Theory to Condensed-Matter Physics, and more generally, Quantum Physics. In particular, we introduce novel quantum algorithms, both variational and exact. We also study the so-called overparametrization phenomenon in VQAs, and the scalability of certain Hamiltonian learning quantum algorithms to a large number of qubits.

The thesis is organized as follows. In Chapter 2, we compute the quantum Fourier transform of the Prime state (a uniform superposition of prime numbers) and show that it provide access to Chebyshev-like biases in the distribution of primes. We also study the entanglement traits of the Prime state and other quantum related states, and find a connection between the entanglement

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18 CHAPTER 1. INTRODUCTION entropy of half chain and the Shannon entropy of the square-free numbers. In Chapter3, we introduce a novel VQA to compute the Schmidt decomposition of an unknown pure bipartite state. Moreover, we show two further applications of this algorithm, including the possibility to swap quantum states without using quantum gates connecting the two parties. In Chapter 4, we provide the first theoretical analysis of the overparametrization phenomenon in VQAs, with several rigorously-proven results. In Chapter5, we convert the algebraic Bethe ansatz, which delivers the eigenstates of a large class of integrable models, into a quantum circuit for its implementation on a quantum computer. In Chapter6, we derive conditions under which quantum Bayesian Hamiltonian learning algorithms may be scalable (for commuting Hamiltonians), and find the optimal evolution time to initialize them. Finally, we conclude in Chapter7 with some remarks and future outlook.

We also provide supplemental material in the form of appendices. In Ap- pendixA, we introduce qibo, an open-source library for the classical simulation of quantum circuits with hardware acceleration. Qibo was employed in all research projects that came after we developed it, namely chapters4,5,6. In Ap- pendixB, we show the proofs for the theoretical results presented in Chapter2.

In AppendixC, we provide the proofs for the theoretical results in Chapter4.

Then, we introduce in Appendix Dsome additional theoretical results to com- plement Chapter5. Lastly, we present the proofs for the results of Chapter6 in AppendixE.

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Chapter 2

The Prime state and its quantum relatives

“There is music in the primes”

Michael Berry

Prime numbers have fascinated mathematicians for millennia, since the times of Euclid in the sixth century BC. Quantum mechanics have intrigued physicists and philosophers alike since its inception at the beginning of the twentieth century. In this chapter, we combine elements from both worlds and present a novel approach to Number Theory using the tools of Quantum Information Theory. This approach, that we call Quantum Number Theory, was started in Ref. [LS14], where the Prime state was introduced, and shown to be efficiently preparable on a quantum computer. Then, in Ref. [LS15] the entanglement entropy of this state was shown numerically to follow a volume law, scaling linearly with system size. Another study that considers quantum states built upon sequences of integers can be found in [MR14].

The Prime state encodes, quantum-mechanically, arithmetic properties of the primes. In this chapter, we prove that the Quantum Fourier Transform (QFT) of the Prime state provides a direct and efficient access to Chebyshev-like biases in the distribution of prime numbers. Furthermore, we analytically characterize the entanglement spectrum of the state in terms of number-theoretic functions, and we find a relation between the scaling of the entanglement entropy and the density of square-free integers. The latter result is stated formally as a conjecture, although we provide theoretical arguments and numerical support in favor of it.

On the numerical side, we show that the scaling of the entanglement entropy of the Prime state is invariant under changes of local dimension, i.e. it remains the same when going from a qubit basis to a qudit basis. This implies that it is an intrinsic property of the distribution of the primes, irrespective of the

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20 CHAPTER 2. THE PRIME STATE AND ITS QUANTUM RELATIVES basis in which these are expressed. We also introduce “quantum relatives” of the Prime state, most notably arithmetic Prime states, which are the quantum superposition of primes in arithmetic progressions. We show that these states have an entanglement entropy that scales precisely as that of the Prime state.

Finally, other states built from the superposition of sequences of integers, like odd composites, square-free numbers and starry primes, are investigated. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.

The chapter is organized as follows. In Sec.2.1, we cover everything related to the Prime state. Namely, we review its definition and efficient construction on a quantum computer, we introduce its QFT, and we study the entanglement present in the state. Then, in Sec. 2.2, we introduce and study the entanglement traits of the quantum relatives of the Prime state. Finally, we provide a discussion and future prospects for Quantum Number Theory in Sec.2.3.

2.1 The Prime state

Mathematical sequences of integers are key to Number Theory and other areas of Mathematics. Properties of such sequences can be studied on a quantum computer by creating pure states consisting of superpositions of computational- basis vectors that encode the numbers appearing in the sequence, e.g. in binary format using qubits. We refer to these states as number-theoretic quantum states. In particular, the quantum state built from the superposition of prime numbers is what we call the Prime state. This state, its efficient construction on a quantum computer, and how it can be used to test the Riemann hypothesis, are reviewed in Sec.2.1.1. In Sec.2.1.2, we compute the QFT of the state, and in Sec.2.1.3we study its entanglement traits.

2.1.1 Efficient construction on a quantum computer

The Prime state |Pⁿi of n qubits is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers written in binary format, up to 2ⁿ (we assume that n ≥ 2). That is,

|Pni ≡ 1 pπ(2ⁿ)

2ⁿ

X

p: prime

|pi , (2.1)

where π(x) is the prime-number counting function, which gives the number of primes smaller than or equal to x, p = pn−12ⁿ⁻¹+ · · · + p12¹+ p02⁰, and

|pi ≡ |pn−1i ⊗ · · · ⊗ |p1i ⊗ |p0i, with pi = {0, 1}. This state was introduced in Ref. [LS14], where it was shown that its construction on a quantum computer is efficient. There are two ways of proceeding. The first simpler method is probabilistic; a second one makes use of Grover’s algorithm [Gro96]. Let us review in more detail these two algorithms for constructing the Prime state.

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2.1. THE PRIME STATE 21 The fundamental element to create a Prime state in either a probabilistic or deterministic manner is to design a unitary operation U_primethat discriminates primes from composites. This unitary acts as follows,

Uprime|ki ⊗ |ai ≡







|ki ⊗ X |ai if k is prime

|ki ⊗ |ai if k is not prime

, (2.2)

where |ki is a n-qubit computational-basis state, |ai is a single ancilla qubit, and the action of the X gate is given by the Pauli matrix σx. Notice that if the state of the ancilla is |ai = ^√¹

2(|0i − |1i), as it occurs in Grover’s subroutine, the action of Uprimeon a superposition is to introduce relative minus sign for prime numbers. The relevant observation here is that the above unitary amounts to code on a quantum computer a primality test, which is a language that belongs to the complexity class P ¹. As a consequence, the operation Uprime

only involves a polynomial number of quantum gates, which will depend on the specific primality test chosen. For the sake of illustration, a detailed explicit form of a quantum primality oracle with O(n⁶) gates, based on the classical Miller-Rabin test [Mil76;Rab80], was produced in Ref. [LS14]. Other primality algorithms, such as the AKS primality test [AKS04], could be used as well.

We can then consider a first probabilistic algorithm to create the Prime state. We need to apply the unitary Uprimeon the uniform superposition of all computational-basis vectors, |φi = ^√¹

2ⁿ

P2ⁿ−1

i=0 |eii, plus an ancilla in the |0i state. Here, the state |φi is created by applying n Hadamard gates, H^⊗n, to the initial |0i^⊗nstate. This yields:

U_prime|φi =

r2ⁿ− π(2ⁿ) 2ⁿ

X

p: not prime

|pi|0i +

rπ(2ⁿ) 2ⁿ

X

p: prime

|pi|1i . (2.3)

By measuring the ancillary qubit in the above state, and post-selecting whenever the result of the measurement is |1i, one obtains the Prime state. This will occur with probability ∼ _{n ln 2}¹ , according to the prime number theorem (see Eq. (2.4)). Because this probability decreases only polynomially with n, this is an efficient method to create the state.

A more refined, deterministic way to create the Prime state consists in using a Grover’s algorithm that uses a primality test as oracle. The efficiency of the algorithm hinges on two facts. The first was mentioned previously, namely, the oracle based on U_primeis efficient. The second fact that guarantees the efficiency in the construction of the Prime state is the relatively high abundance of primes, π(x), which is asymptotically given by the logarithmic integral function, Li(x) ≡ Rx

2 dt ln t, i.e.

π(x) ∼ Li(x) −^x→∞−−−→ x

ln x . (2.4)

1Informally, the complexity class P is the class of decision problems that can be solved by a classical computer within a time that scales polynomially with the size of the input.

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22 CHAPTER 2. THE PRIME STATE AND ITS QUANTUM RELATIVES This result is known as the Prime Number Theorem (PNT) [Apo76], and it implies that the needed number of calls to Grover’s oracle is only O(√

n). Indeed, this estimate is given by ^π₄q

N

S [NC10], where N = 2ⁿ is the dimension of the Hilbert space of n qubits and S is the number of solutions to the oracle, i.e.

S = π(2ⁿ). Therefore, the overall computational complexity of generating a Prime state of n qubits on a quantum computer with this method, using e.g.

Miller-Rabin primality test, is O(n^6.5).

Once an oracle for the Prime state has been constructed, it can be used to numerically verify (or falsify) the Riemann hypothesis [Rie59; Edw74], which constitutes one of Clay Mathematics Institute’s Millennium Problems [Ins00].

The Riemann hypothesis states, for the distribution of prime numbers, that the deviations of π(x) from Li(x) are bounded for large x as [Sch76]

|π(x) − Li(x)| ≤ 1 8π

√x ln x for x > 2657 . (2.5)

The quantum algorithm proposed in Ref. [LS14] may allow to compute π(x) beyond the limits achievable by classical supercomputers –once a fault-tolerant universal quantum computer becomes available– by using a quantum counting algorithm that provides an estimate of the number of terms in superposition in the Prime state. So far, π(x) have been computed up to 10²⁷ [WB15], which implies that a Prime state with a minimum of ∼ 90 logical qubits would be needed to surpass this computation (since 2⁹⁰' 1.238 × 10²⁷).

To do so, it was suggested to apply a quantum counting algorithm [BHT98]

that delivers the number of solutions S to an oracle search problem; in this case, the number of primes that are marked by Grover’s oracle, G. This algorithm uses Quantum Phase Estimation (QPE) [Kit95] to obtain the eigenvalues of G, which in turn reveal the number of solutions to the query problem. In order to obtain an estimate of π(x) that is meaningful, the precision in the estimation should be smaller than the fluctuations allowed by the Riemann Hypothesis. To achieve such precision, O(√

x) calls to the oracle are needed, with x = 2ⁿ. This is quadratically better than the performance of any classical counterpart in an identical oracular setting of the counting problem. Recently, there have been new proposals for quantum counting the solutions to an oracle G that do not rely on QPE [Aar19;AR20]. In practice, any quantum counting algorithm may be applied.

For the sake of completeness, let us recall that the best classical algorithms that compute π(x) unconditionally, i.e. the validity of the estimation not depending on the truthfulness of the Riemann hypothesis or any other unproven conjecture, do not however rely on enumerating all primes. The main algorithm for computing π(x) is a combinatorial method due to Meissel [Mei70] and Lehmer [Leh59], which has subsequently been improved by Lagarias, Miller and Odlyzko [LMO85], Del´eglise and Rivat [DR96], and Gourdon [Gou01]. The latter version of this algorithm, which has time complexity O(x^2/3ln⁻²x), was used to compute π(10²⁷), the world record. Hence, quantum counting on the Prime state provides a polynomial speed-up over this method. Another two analytic

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2.1. THE PRIME STATE 23 algorithms for computing π(x), put forward by Lagarias and Odlyzko [LO87], have time complexity O(x^1/2+) and O(x^3/5+) respectively, with > 0.

2.1.2 Quantum Fourier transform

The Quantum Fourier Transform (QFT) is a unitary operation that plays a key role in several important algorithms, such as Quantum Phase Estimation [Kit95]

or Shor’s algorithm [Sho94]. Its action on a quantum state |ψi =P

jx_j|eji of n qubits, expressed in the computational basis, {|e_ji}, is given by

UQF T|ψi =

2ⁿ−1

X

k=0

yk|eki , yk≡ 1

√2ⁿ

2ⁿ−1

X

j=0

xje^{2πi jk/2}ⁿ , (2.6)

where the {yk} are the discrete Fourier transform of the original amplitudes {xj}, and the i in the exponent is the imaginary unit. The QFT is an efficient subroutine, requiring O(n²) gates on a quantum computer [NC10].

In the present work, we calculate the QFT of the Prime state, |ˆPni ≡ UQF T|Pni, that is,

|ˆPni = 1 p2ⁿπ(2ⁿ)

2ⁿ−1

X

k=0



 X

p: prime

e^{2πi pk/2}ⁿ



|ki . (2.7)

Notice that the amplitudes in Eq. (2.7) are symmetric with respect to the central value k = N/2, where N = 2ⁿ. This means that the probability of measuring the state |ki and |2ⁿ− ki is the same. Indeed:

P (2ⁿ− k) = 1 2ⁿπ(2ⁿ)

X

p: prime

e^{2πi p}e^{−2πi pk/2}ⁿ

2

= 1

2ⁿπ(2ⁿ)

X

p: prime

e^{−2πi pk/2}ⁿ

2

= P (k) .

(2.8)

This is a general property of the QFT of a state with real amplitudes, that is verified by the Prime state, see Fig. 2.1.

The main result of the computation of |ˆPni is that the QFT applied to the Prime state provides an efficient method for estimating Chebyshev-like biases in the distribution of prime numbers. These biases reflect the unbalance in the number of primes, below a certain value, appearing in different arithmetic progressions.

Let us therefore consider arithmetic progressions of the form αk + β, for k = 0, 1, 2, . . . , with α, β coprimes (i.e. gcd(α, β) = 1). Dirichlet proved that there exists an infinite number of primes in any of these sequences [Apo76].

The number of primes in the sequence αk + β (with α, β coprimes), below a

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24 CHAPTER 2. THE PRIME STATE AND ITS QUANTUM RELATIVES

0 5000 10000 15000 20000 25000 30000

k

0.00 0.02 0.04 0.06 0.08 0.10

P (k)

Quantum Fourier Transform of the Prime State

Figure 2.1: Numerical simulation of the Quantum Fourier Transform of a Prime state with n = 15 qubits. The probability peaks appear symmetrically distributed with respect to the central value k = 2ⁿ/2. Note as well that there is no peak for the value k = 2ⁿ, because the state |2ⁿi is not included in the Hilbert space of n qubits.

certain value x, is denoted by the modular prime counting function π_α,β(x), in close analogy to the prime counting function, π(x). The asymptotic behaviour of these modular counting functions is determined by

π_α,β(x) ∼ 1

φ(α)Li(x) −^x→∞−−−→ 1 φ(α)

x

ln x , (2.9)

where φ(α) is the Euler’s totient function, which gives the number of coprimes to α smaller than α. For example, if p is a prime, one has φ(p) = p − 1, because all the integers below p do not divide it. Notice as well that φ(α) is the number of arithmetic progressions of the form αk + β that can contain primes, which are those in which β is coprime to α. Thus, Eq. (2.9) means that the number of primes is, on average, asymptotically equi-distributed among the existing progressions αk + β (α, β coprimes), for fixed α. Nevertheless, when considering a finite sequence of primes, there exists biases in the distribution of the primes among different progressions that are quantified by the functions

∆α; β₁,β₂(x) ≡ πα,β₁(x) − πα,β₂(x) . (2.10)

These biases can be numerically estimated, to a desired additive precision , from the probability peaks appearing in |ˆPni. In particular, relevant peaks appear at values k = N/3, N/4, N/6, and their mirror images about the central peak (see App.B.1for a derivation of these results). The expressions for these

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2.1. THE PRIME STATE 25 peaks are,

P (N/3) = ∆_3;2,1(N )²+ π_3,1(N )π_3,2(N ) − π(N ) + 2

N π(N ) , (2.11)

P (N/4) = 1 + ∆(N )²

N π(N ) , (2.12) P (N/6) = ∆6;5,1(N )²+ π6,1(N )π6,5(N ) − 3 π6,1(N ) + 3

N π(N ) , (2.13)

where ∆(N ) ≡ π4,3(N ) − π4,1(N ) is the Chebyshev bias. There exists a direct relation between the functions π3,1(x), π3,2(x) and π6,1(N ), π6,5(N ). In fact, π3,1(x) = π6,1(x) and π3,2(x) = π6,5(x) + 1. This is so because every prime p counted by π_3,1(x) is of the form p = 3k + 1, but since p is odd, k must be even (k = 2k⁰), and p = 6k⁰+ 1, so π_3,1(x) = π_6,1(x). On the other hand, every prime p counted by π_3,2(x) is of the form p = 3k + 2, with k odd (k = 2k⁰+ 1), so p = 6k⁰+ 5; in this case, 2 is the only prime counted by π_3,2(x) and not by π_6,5, and thus π_3,2(x) = π_6,5(x) + 1. These relations,

π_3,1(x) = π_6,1(x) , π_3,2(x) = π_6,5(x) + 1 , (2.14) allow to obtain π3,1(x), π3,2(x), π6,1(N ), π6,5(N ), ∆3; 2,1(N ) and ∆6; 5,1(N ), by substituting Eq. (2.14) into the probability peaks P (N/3) and P (N/6) in Eqs. (2.11) and (2.13), and solving the resulting system of two equations and two unknowns. Of course, π(N ) must be known in advance, but this is already provided by the Prime state combined with some quantum counting algorithm.

Notice as well that N/3 and N/6 do not correspond to integer values for qubit systems, where N = 2ⁿ. This means that the associated peaks cannot be directly measured in the computational basis. Instead, one needs to resort to ancillary qubits to represent rational numbers as binary fractions; the QFT acts upon the two registers, the one representing the integer part and the one representing the fraction. A numerical check of the error incurred in the estimation of the peak when using nine ancillary qubits to represent the non-integer part of N/3 or N/6 is shown in Fig.2.2, for n = 10-30 qubits. The relative error is around 10⁻⁶ for 30 qubits. If more precision is required, one can simply add more ancillas.

It is important to consider how efficient it is to measure the peaks given in Eqs. (2.11), (2.12) and (2.13), to a desired precision, when n grows large. Note that P (N/3) and P (N/6) are dominated by the cross product of the modular prime counting functions, divided by N π(N ). According to Eq. (2.4) and Eq. (2.9), these probabilities will then decrease as ∼ _φ(α)¹ ₂_{n ln 2}¹ , where α = 3, 6 respectively. This decrease is only polynomial in n, and therefore, the method is efficient. This is so because the number of measurements required to achieve a target precision in the estimation of outcome probabilities, {pi}, is dictated by the sampling process of a multinomial distribution. In this context, the statistical additive errors ², {i}, arise from finite sampling, and depend on the

2If pi is the exact probability, and ˆpi the estimate of it, the statistical additive error is defined as pi= ˆpi+ i, and the multiplicative error as ˆpi= (1 + εi)pi.

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10 12 14 16 18 20 22 24 26 28 30

n

0.000 0.001 0.002 0.003 0.004 0.005

relative error (%)

Relative error in the estimation of peaks

N/3N/6

Figure 2.2: Relative percentage error in the estimation of Eqs. (2.11) and (2.13) due to the limited precision in the representation of N/3 and N/6, using nine ancillary qubits, for n = 10 − 30 qubits.

number of measurements performed, #M , as _i =q_p

i(1−p_i)

#M . Hence, in order to achieve a small multiplicative error εi, say εi≈ 0 and i<< pi, then the number of measurements, or runs of the quantum circuit, must be #M >> ^1−p_p ⁱ

i . Therefore, it is clear that estimating probabilities with a small multiplicative error remains efficient as long as these probabilities show at most a polynomial decrease with increasing n.

Moreover, direct application of the QFT on the Prime state, followed by the procedure of Amplitude Estimation [Bra+02], allows to obtain a dependency of the additive error on the number of runs of the quantum circuit that is O(_#M¹ ), i.e. a quadratic improvement compared to direct sampling from the multinomial distribution. This is the same technique employed for instance in the quantum Monte Carlo algorithm [Mon15]. In this case, all it takes to prepare |ˆPni for Amplitude Estimation is to apply a single n+a+1-qubit (where a is the number of ancillas used for the non-integer part) controlled-X gate on an extra ancilla qubit, that singles out the desired peak, e.g. |N/3i,

N −1

X

k=0 k6=N/3

yk|eki|0i +p

P (N/3) |N/3i|1i . (2.15)

States like the one in Eq. (2.15) are ready for Amplitude Estimation. Clas- sically, the best algorithm for computing modular prime counting functions, πα,β(x), is a variation of the Meisser-Lehmer method for computing π(x), and has time complexity O(x^2/3ln⁻²x) [DDR04]. To the best of our knowledge, modular prime counting functions have been computed for values only up to x = 10⁹[Wei]. Hence, a quantum computer could be helpful in this sort of com-

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2.1. THE PRIME STATE 27 putations and provide a polynomial speed-up over classical methods, depending on the precision required in the estimation. That is, if the additive precision is lower bounded as Ω(2^−pn), then the sample complexity (i.e. the number of copies of the state in Eq. (2.15)) is O(2^pn). If p < 2/3, a polynomial speed-up is achieved.

In contrast to P (N/3) and P (N/6), the peak P (N/4), which delivers the Chebyshev bias, ∆(N ), is unlikely to be of practical use because the denomi- nator N π(N ) is expected to cause an exponential decrease in this probability, as the number of qubits n increases. As already explained, this would demand an exponentially large number of measurements. Yet, it is interesting to appre- ciate how the amplitudes of the Prime state interfere under the QFT to give some remarkable meaning, in terms of number-theoretic functions, to certain values of k. Note as well that if one is willing to directly sample from |ˆPni, all the information contained in the probability peaks is extracted simultaneously, rather than sequentially, by accumulation of statistical knowledge on measurement outcomes. This means that the values of all peaks are estimated altogether with a fixed number of samples, to a desired additive precision .

It also happens that the QFT of an equally-weighted superposition provides a means to count the number of terms in the computational basis. Indeed, the probability of finding the |0i^⊗n state on a measurement after the application of the QFT equals S/N , where again N = 2ⁿ is the dimension of the Hilbert space of n qubits and S is the number of terms in the superposition (in the computational basis). A derivation of this result can be found in App.B.2. For

|ˆPni, this probability reads

P (0) = π(N )

N , (2.16)

which decreases as ∼ _{n ln 2}¹ in the limit n 1, according to the PNT, Eq. (2.4).

The prime counting function, π(N ), appears as well in the central peak, where k = N/2, as shown in App.B.1. The expression for this peak is

P (N/2) = π(N )²− 4π(N ) + 4

N π(N ) , (2.17)

which also decreases linearly in n⁻¹.

Because the precision required to meaningfully test Riemann’s hypothesis, Eq. (2.5), implies that the multiplicative error must become exponentially small for increasing n (as

√x ln x

π(x) decreases as O n²2^−n/2), direct sampling from |ˆPni is not competitive with previously mentioned quantum counting algorithms, requiring O(2ⁿn⁻³) repetitions. On the other hand, it is possible to prepare a state like the one in Eq. (2.15), but marking the peak P (0) instead, and then apply Amplitude Estimation. However, note that if one has an oracle, it is possible to generate, with a single query to that oracle, the state in Eq. (2.3).

Such state is also ready for AE, and can be used to estimate π(N ) = S/N , but it is easier to prepare than the one in Eq. (2.15). So there is no point in using

|ˆPni for estimating π(2ⁿ).

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28 CHAPTER 2. THE PRIME STATE AND ITS QUANTUM RELATIVES The QFT may nonetheless be useful to estimate the number of terms in an equally-weighted superposition, for states for which one does not know of any oracle; for instance, a ground state of a Hamiltonian obtained from a variational algorithm such as the Variational Quantum Eigensolver [Per+14] or the Quantum Approximate Optimization Algorithm [FGG14].

2.1.3 Entanglement traits

Some relevant properties of the prime sequence (and presumably other sequences of numbers and their corresponding quantum states) are encoded into the density matrix and the quantum correlations among partitions of the Prime state, and are therefore amenable to investigation using a quantum computer. It is specially interesting to note that the Prime state bears an amount of entanglement that is almost maximal across many, if not all, bi-partitions (as characterized by e.g. the von Neumann entropy [LS15]). Very large entanglement is tantamount to large quantum correlations related to non-locality. As a consequence, the Prime state is genuinely quantum, that is, it encodes correlations that cannot be described in classical terms.

The main interest in studying the entanglement traits of the Prime state, and other number-theoretic quantum states, is that it is not at all unreasonable to think that its quantum correlations may be related to deep facts in Number Theory. For instance, the Hardy-Littlewood constants [HL23] that characterize pairwise correlations among primes appear in the asymptotic expression of the reduced density matrix of the Prime state, for natural bi-partitions³. This fact and others that we shall present in this work can be interpreted as convinc- ing evidence supporting the aforementioned statement. Quantum entanglement may indeed help unravel deep truths in Number Theory.

The entanglement properties of quantum states can be studied by taking different bi-partitions of the states and quantifying their entanglement, with figures of merit such as the von Neumann entropy, the purity or the entanglement spectrum, among many possibilities. Unfortunately, a practical way to quantify genuine multipartite entanglement, not related to partitions of the system, is not available for states with a large number of qubits [B ˙Z06].

As already mentioned, the entanglement present in number-theoretic quantum states is the result of correlations among the digits of the numbers appearing in the superposition. In turn, these observables may reveal information about pairwise correlations between the numbers in the sequence. The reduced density matrix of the Prime state upon taking a natural bi-partition A–B with the m rightmost qubits, ρ_A, is asymptotically given by ρ_A[LS15], whose expression is

ρ_A= 1

d(1 + `NC_m) , (2.18)

where d = 2^m−1, `N ≡ ^Li_{Li(N )}²^{(N )} −−−−→^{N →∞} _{n ln 2}¹ , and Cm is a d × d symmetric,

3We consider natural bi-partitions those that separate the first m qubits and the remainder n − m qubits, without any reordering.

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2.1. THE PRIME STATE 29 traceless Toeplitz matrix defined by

(C_m)_i,j≡ (1 − δ_i,j) C(2|i − j|), i, j = 1, . . . , d , (2.19) where C(h) are the Hardy-Littlewood constants, defined by

C(h) =







2 C2Q

p>2; p|h p−1

p−2 if h is even

0 otherwise

, (2.20)

with C₂=Q

p>2

1 − _(p−1¹₂₎

the twin prime constant and {p} the prime numbers. The constants C(h) characterize the pairwise correlations among prime numbers, and it is precisely due to these correlations that the reduced density matrix ρ_A is not maximally mixed. To be precise, the first Hardy-Littlewood conjecture [HL23] assures that the number of prime pairs (p, p + k) up to a certain number x, i.e. π_k(x), is given by

πk(x) ∼ C(k) Li2(x) −−−−→ C(k)^x→∞ x

(ln x)² , (2.21) where Li2(x) =Rx

2 dt (ln t)².

An analytical approximation to the positive eigenvalues of the matrix C_m was conjectured in Ref. [LS15]. That approximation was suitable for the highest eigenvalues, but failed to describe the lowest ones. Moreover, the degeneracies of the former failed beyond some point⁴. A recent article gives a heuristic derivation of the Hardy-Littlewood conjecture (2.21) starting from the pair correlation formula for the Riemann zeros, including lower order terms [KS19]. The reverse statement was known since long and played an important role in the connection between Number Theory and Quantum Chaos [BK96; BK99]. The derivation of Ref. [KS19] employs the following expression for the Hardy-Littlewood constants,

C(h) =

∞

X

k=1

µ(k) φ(k)

2

ck(h) , (2.22)

where φ(x) is the Euler’s totient function, µ(x) is the M¨obius function (see below), and

ck(h) =

k

X

l=1 gcd(l,k)=1

e^{2πi hl/k} (2.23)

are the Ramanujan’s sums. Equation (2.22) is the Ramanujan-Fourier series of the constants C(h) and has also appeared in the context of random processes, more concretely, in the relation between the power spectrum and the correlation function using the Wiener-Khintchine formula [GP99;GP06].

4We thank Alonso Botero for noticing this fact.

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0 10 20 30 40 50 60 70 80

3 4 5 6 7 8

k

(1) = 1 (3) = 2

(5) = 4 (7) = 6

(15) = 8 (11) = 10 (13) + (21) = 24 (17) = 16

(1) = 1 (3) = 2

(5) = 4 (7) = 6

(15) = 8 (11) = 10 (13) + (21) = 24 (17) = 16

Entanglement spectrum of the Prime State

Exact Approximated

Figure 2.3: Comparison of the exact lowest 72 entanglement energies of the entanglement spectrum of the Prime state to those computed using Eq. (2.25), for n = 30 qubits. We expect this approximation to improve for n & 64, based on the very good approximation obtained for the slope of the entanglement entropy (Eq. (2.31)) shown in Fig.B.3(right)in App.B.3.

Based on Eq. (2.22), a more precise description of the eigenvalues of C_m can be obtained (for a justification see App. B.3). To every square-free odd integer k, that is, to every odd integer k such that it does not contain any prime raised to a power larger than one in its prime decomposition, we associate an eigenvalue γk of Cmwith degeneracy φ(k). These eigenvalues are given by

γ_k ' 2^mµ²(k)

1 φ²(k)− 1

φm

, k = 1, 3, 5, . . . , k_m, (2.24)

where the M¨obius function, µ(x), takes values ±1 for square-free numbers and 0 otherwise. The values of kmand φm are fixed by the dimension and normal- ization of the density matrix in Eq. (2.18), and kmscales as 2^m/2(see App.B.3 for details). There are cases where the eigenvalue is the same for two different values of k, k⁰, in which case the degeneracy is higher. An example of such an accidental degeneracy is bγ₁₃ = bγ₂₁ = ₁₄₄¹ , which has a total degeneracy φ(13) + φ(21) = 24 (we denote by bγ_k = µ²(k)/φ²(k)). Another example is bγ₃₅=bγ₃₉=₅₇₆¹ , that has a degeneracy φ(35) + φ(39) = 48.

Using Eq. (2.24), the eigenvalues of the density matrix (2.18) are approximated by

λk' 2^1−mµ²(k)

1 + 2^m n ln 2

1

φ²(k)− 1 φ_m

, (2.25)

with a degeneracy φ(k). The good agreement between the exact eigenvalues and those computed from Eq. (2.25) for n = 30 qubits, is shown in Fig. 2.3, where the lowest 72 entanglement energies of the entanglement spectrum are

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2.1. THE PRIME STATE 31

8 10 12 14 16 18 20 22 24 26 28 30 32 n

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Ratio to analytic formula

Trace of Csm

s = 2 s = 3 s = 4 s = 5

22 24 26 28 30 32 0.980

0.985 0.990 0.995 1.000

8 10 12 14 16 18 20 22 24 26 28 30 32 n

7 6 5 4 3 2 1 0

ln (1-ratio to analytic formula)

Exponential convergence of Trace Csm s = 2 s = 3 s = 4 s = 5

Figure 2.4: left) Ratio of Tr (C^sm) to Eq. (2.27), for s = 2, 3, 4, 5 and m = ⁿ₂, up to n = 32 qubits. right) Natural logarithm of (1− ratio of Tr (C^s_m) to Eq. (2.27)), up to n = 32 qubits. An exponential convergence is observed as the number of qubit increases.

plotted. These correspond to the highest eigenvalues of the density matrix of the Prime state, since the entanglement spectrum {εk} of a density matrix ρ with eigenvalues {λk} is defined as

εk= − ln λk. (2.26)

Using again Eq. (2.24), we can estimate the contribution of the positive eigenvalues to the trace of the powers of C_m, from which all R´enyi entropies can be readily derived [R´en61]. The computation is as follows,

Tr C^s_m '

k_m

X

k=1 (odd)

φ(k) γ^s_k

−−−−→m1 2^ms

∞

X

k=1 (odd)

|µ(k)|

(φ(k))^2s−1

= 2^msY

p>2

1 + |µ(p)|

(φ(p))^2s−1

= 2^msY

p>2

1 + 1

(p − 1)^2s−1

, (2.27)

where we have used that for any function f (k) which is multiplicative (see App.B.3), it holds that

∞

X

k=1 (odd)

|µ(k)| f (k) = Y

p>2

(1 + f (p)) , (2.28)

where the product runs over the odd prime numbers. We have also used that φ(p) = p − 1 for prime numbers, and that µ(k)² = |µ(k)|. In Eq. (2.27) we

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6 8 10 12 14 16 18 20 22 24 26 28 30 n

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

c

H(

32)

Value of c

14 16 18 20 22 24 26 28 30 0.875

0.880 0.885

0.890 H(

32)

6 8 10 12 14 16 18 20 22 24 26 28 30 n

0.5 0.0 0.5 1.0

1.5 1 +³2

Value of

14 16 18 20 22 24 26 28 30 1.225

1.250 1.275 1.300

1.325 1 +³2

Figure 2.5: left) Value of cπ(n) computed according to Eq. (2.31), compared to the conjectured value H _π³₂ (horizontal black line), up to n = 30 qubits; right) Value of γ(n) computed according to Eq. (2.32), up to n = 30 qubits, compared to the conjectured value 1 +_π³2 (horizontal black line).

dropped the negative term proportional to 1/φmof Eq. (2.24) because its contribution vanishes in the limit N, m 1 for s > 1. For s = 1, the Riemann zeta function ζ(1) arises and the product diverges because we have dropped the negative eigenvalues of the matrix Cm. For s = 2, Eq. (2.27) coincides with the heuristically-derived asymptotic expansion of Tr C²_min Ref. [LS15]; furthermore, this equation was conjectured to hold for any s ≥ 2. Here, we proved that it follows from our analytical approximation to the eigenvalues of the reduced density matrix, Eq. (2.24). Figure2.4shows the ratio of the exact value of the trace of the matrix C^s_m to the asymptotic analytical formula in Eq. (2.27), and the exponential convergence of this ratio towards one, up to n = 32 qubits.

These results indicate that the contribution of the negative eigenvalues of Cm

to the trace of the powers of this matrix is asymptotically negligible, for s ≥ 2.

In contrast, their contribution to the von Neumann entropy (s = 1) does not appear to be so.

The von Neumann entropy S (also known as entantlement entropy) of a density matrix ρ, with eigenvalues {λi}, is defined by

S = −Tr (ρ log₂ρ) = −X

i

λilog₂λi , (2.29) and it is a relevant measure of bipartite entanglement. The von Neumann entropy of the reduced density matrices of equal-sized bi-partitions of the Prime state, with an even number of qubits, was numerically calculated in Ref. [LS15], and found to scale linearly with the size of the bi-partition, thus following a

“volume law”. To be precise, the best fit to a line, for n = 20 – 30 qubits and for the natural equal-sized bi-partition, is S(n) = 0.88612902ⁿ₂− 1.30405956. This result indicates that the Prime state is highly entangled, but not maximally so, because the maximal possible scaling for the entropy would be linear in

n

2 with a slope equal to 1. Random states, i.e. those with random complex coefficients, also follow a volume law, but with an entanglement entropy of half chain that scales maximally as ⁿ₂ − 1/2, for big n [Pag93]. If we restrict to real

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2.1. THE PRIME STATE 33

6 8 10 12 14 16 18 20 22 24 26 28 30

n

8 6 4 2 0

ln

Exponential convergence of c and c

Figure 2.6: Natural logarithm of the absolute value of the difference δ, between the observed and predicted values of cπ and γ, up to n = 30 qubits. That is, δ ≡ cπ(n) − H _π³2

and δ ≡

γ(n) − 1 +_π³2

, respectively. An exponential decrease in δ is observed as the number of qubit increases, before reaching a plateau.

positive random coefficients, the scaling is (1 − 2/π)ⁿ₂ = 0.363 . . .ⁿ₂ [GF15]. So the Prime state is not a typical random positive state either. Notice as well that the high entanglement of the Prime state implies that it is not possible to apply standard classical techniques such as Matrix Product States (MPS), or other more-refined tensor networks, to efficiently simulate the Prime state on a classical computer [Lat07].

We herein conjecture that the entropy of the natural equal-sized bi-partition of the Prime state of n qubits is asymptotically given by

S(n) = c_π n

2 − γ , c_π = H 3 π²

, γ = 1 + 3

π² , (2.30) where H(p) ≡ −p log₂(p) − (1 − p) log₂(1 − p) is the Shannon entropy and 3/π²= 1/(2 ζ(2)) is equal to a half of the asymptotic density of odd square-free integers [Apo76]. The constant 3/π² also appears in the study of topological dynamical systems. In particular, it has been shown that it gives half the topological entropy of the square-free flow [Sar11; SC13]. The term -1 in the intercept comes from the fact that the first or least relevant qubit is basically in the state |1i, because all primes but 2 are odd, and hence this qubit does not contribute to the entropy in the asymptotic limit. In order to test the conjectured Eq. (2.30), we have computed the entropy of the Prime state up to n = 30 qubits. Diagonalization using quadruple-precision floating-point numbers was implemented to meet the precision demanded, for large values of n, by the observed oscillatory behaviour of the slope c_π. To the best of our knowledge, this is the first open-source library that diagonalizes matrices using float128; as a

Quantum algorithms for condensed-matter physics, number theory and quantum machine learning

Quantum Algorithms for Condensed-Matter Physics, Number Theory and Quantum

Machine Learning

Diego Garc´ıa Mart´ın

Supervised by:

Germ´ an Sierra Rodero Jos´ e Ignacio Latorre Sent´ıs

Ph.D. thesis

September, 2022

Quantum Algorithms for Condensed-Matter Physics, Number Theory and Quantum

Machine Learning

Diego Garc´ıa Mart´ın

Programa de Doctorado en F´ısica Te´ orica Departamento de F´ısica Te´ orica

Facultad de Ciencias

Directores: Germ´ an Sierra Rodero

Jos´ e Ignacio Latorre Sent´ıs

Tutora: Francesca Maria Marchetti

Acknowledgments

List of publications

Peer-reviewed

ArXiv preprints

Open-source software

Abstract

Resumen

Contents

Chapter 1

Introduction

Chapter 2

The Prime state and its quantum relatives

2.1 The Prime state

2.1.1 Efficient construction on a quantum computer

2.1.2 Quantum Fourier transform

k

P (k)

Quantum Fourier Transform of the Prime State

n

relative error (%)

Relative error in the estimation of peaks

2.1.3 Entanglement traits

Entanglement spectrum of the Prime State

n

ln

Exponential convergence of c and c