Quantum Digital Cooling

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Quantum Digital Cooling

THESIS

submitted in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

PHYSICS

Author : Stefano Polla

Student ID : 1894056

Supervisor : T. E. O’Brien

In collaboration with : Y. Herasymenko

2ndcorrector : C. J. W. Beenakker

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Quantum Digital Cooling

Stefano Polla

Instituut-Lorentz, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands

June 11, 2019

Abstract

The preparation of a qubit register in the ground state of a given Hamiltonian is a challenging problem in the field of quantum algorithms. Its solution is relevant to enable the study of chemistry, condensed matter and nuclear physics models on a quantum computer Inspired by the way natural systems can be driven to a low energy and low entropy state by

coupling them to a cold bath, we show how a single ancilla qubit periodically measured and reset can be used to drive a system towards

the ground state of an Hamiltonian simulated on a digital quantum computer. We identify caveats that might compromise this method, like violation of the energy conservation principle and non-ergodicity caused

by symmetries, and study strategies to circumvent them. We define and optimize two implementations of an elementary de-excitation procedures

based on the simulation of Hamiltonian evolution and a single non-unitrary operation on the ancilla. By iteration of this procedure, we

construct two protocols that can prepare approximate ground states of composite qubit systems. Using results of numerical simulations, we show that one protocol can prepare an arbitrary-fidelity approximation of the ground state a system of non-interacting qubits with random energies

in polynomial time. The second protocol is designed to have a small computational cost, with the aim of being experimentally realizable in small near-term quantum computers. Both protocols are demonstrated to

succeed in preparing approximate ground states of interacting spin-1/2 chain with transverse-field Ising Hamiltonians. We propose that the methods studied in this thesis can be extended and applied to develop a

novel class of non-unitary quantum algorithms based on ancilla-mediated non-unitary operations

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Chapter

1

Introduction

In 1980 Benioff published the first attempt to construct a quantum me-chanical model of computers [1]. Just the year after, Feynman suggested that natural phenomena that are intrinsically quantum could be simulated efficiently on computers that would store and process information accord-ing to quantum principles [2]. The followaccord-ing years saw the birth and de-velopment of the physics of quantum information and computation. The impact of the field became widely clear in 1994, with Shor’s theorization of the factoring algorithm [3], first showing that quantum computers could be used to solve classically intractable problems. In 1996 Lloyd [4] showed that Feynman’s suggestion was well-founded: one may simulate, to arbi-trary precision, the dynamics of any natural – meaning local – quantum system on a universal [5, 6] quantum computer.

In the same article [4], Lloyd introduced the problem of how to ini-tialize the simulator in a specific state (e.g. a thermal equilibrium state, a natural state for many systems that can exchange energy with an environ-ment). He suggested that the dissipative dynamics through which natural systems cool down could be made itself part of the simulation. Terhal and Di Vincenzo expanded on this concept [7], proving that a thermal state can be prepared in a simulator in the long time limit, by extending the system simulation to include a large heat bath, weakly coupled to the system. The bath can be carefully constructed so that its initial state would be easy to prepare. They defined an equilibration algorithm based on this principles, that can prepare a simulated system in a thermal state, requiring a large time and space overhead to simulate the bath.

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2 Introduction

1.1 Digital quantum simulation

Simulations allow us to obtain information on some model of a phenomenon, by replicating the key elements of the model in a new system rather than observing the phenomenon itself in its original setting. Controlled sys-tem described by models analogous to the one we want to study, allow to perform simulations in the lab. These systems often process informa-tion stored in continuous – or analog – variables. Examples are classical electronic circuits or quantum gases of interacting atoms, which evolve according to equation that map directly to the model of the phenomenon to be simulated. These systems are called analog (quantum) simulators [8]. Analog simulators are typically specialized devices, used to solve a set of problems and that are difficult to re-purpose. Alternatively, models of classical phenomena can be simulated on a digital computer, that works by storing information as binary digits and processing them through a lim-ited set of logic gates. In the same way, we can implement simulations of quantum systems on devices that store information in two-level sys-tems (qubits) and process them by applying sequentially unitary physical transformations taken from some specific set (quantum gates). A device that can perform arbitrary operations on a register of qubits is called uni-versal digital quantum computer [5], and a programmable device that can implement some small set of gates acting on one or two qubits at a time is proven to be universal [6]. Arbitrary digital quantum simulations can be implemented on such a device [4]. Besides direct analog mapping and gate-based digital computing, other quantum simulation and computa-tion methods have been since developed. Among these are the popular adiabatic quantum computing methods, first proposed for function op-timization [9], have since been proven to map to standard computation [10]. Other interesting non-conventional quantum computing principles include dissipation driven quantum computing [11] and measurement-based quantum computing [12].

The advantage of the digital approach lies in the greater level of con-trol that can be generally achieved, the possibility of implementing error-resistant systems [13] (either by physical realization [14] or by active error correction [15]), and the ease of implementing algorithms that go beyond simulation. Digital quantum computers can realize any classical compu-tation [5], and any analog simulation can be approximated by a digital algorithm to arbitrary precision [4]. Besides this, many specific algorithms thought for gate-based digital quantum computers have been developed [16–19].

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1.1 Digital quantum simulation 3

1.1.1 Computational complexity

The greatest success of quantum computing to date is arguably the ex-istence of algorithms that benefit from an exponential quantum speedup [17]. Classes of problems exist, for which a quantum algorithm can find solutions with time and space requirements that are bounded by a polyno-mial in the size of the input, while it’s conjectured that a classical computer would need exponential resources for the same task.

In complexity theory, the class of problems solvable in polynomial time on a quantum computer is called Bounded-error1 Quantum Polynomial (BQP), analogous to the classical Polynomial (P) complexity class. The aforementioned conjecture (compactly written P ( BQP), implies that ∃X problem with variable input size, such that X ∈ BQP∧X /∈ P. A renowned example of X would be the factorization in prime numbers, for which the best known classical algorithm takes exponential time, and for which Shor’s algorithm [3] provides a polynomial-time quantum so-lution. Other classes of problems relevant for us, wider than P and BQP respectively are defined in complexity theory [20]. If an algorithm that can verify whether the problem is solved if given both the input and the output, and this algorithm takes polynomial time on a classical computer, the problem is Nondeterministic Polynomial (NP). If the same is true for an algorithm that runs on a quantum computer, the problem is defined Quantum Merlin-Arthur2(QMA). As solving a problem is enough to ver-ify any output P⊆NP and BQP⊆QMA, and as a quantum computer can implement classical computation P⊆BQP and NP⊆QMA. Finally, there exists the classes of “hardest problems” in QMA and in NP, called QMA-complete and NP-QMA-complete respectively. These contain the problems to which all other problems in QMA or NP can be mapped, by running some mapping algorithm requiring polynomial resources. Panel (a) of Figure 1.1 shows the taxonomy of complexity classes according to current con-jectures. Only non-strict inclusion relations are proven, so some sets may coincide and/or intersect with others.

P and BQP are considered the classes of feasible problems on classi-cal and quantum computers respectively. Finding a quantum polyno-mial time solution any QMA-complete problem would mean that all QMA

1_{A probability of giving the wrong result is allowed to quantum algorithms, as long}

as it’s bounded regardless of the input size, because of the non-deterministic nature of measurement needed to perform the readout.

2_{This name mirrors the one of a probabilistic complexity class (MA), which definition}

is outside the scope of this text. As for BQP, the algorithm that performs verification is allowed to give false positives or negatives with a bounded probability.

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4 Introduction (a) QMA NP BQP P QMA compl. NP compl. (b) QMA=BQP NP P QMA compl. NP compl.

Figure 1.1: Venn diagram of possible inclusion relations between complexity classes. Red arrows show some examples of possible mappings of a problem into another that can be performed in polynomial time. (a) Known and currently-conjectured relations. (b) How relations between complexity classes would be revised if a QMA-complete problem that can be solved in polynomial-time on a quantum computer is found.

problems can be mapped to it and thus solved in polynomial time, QMA= BQP ⊇ NP. This would imply that all problems that can be feasibly ver-ified (NP) could also be feasibly solved with a quantum computer, in op-position to what is currently conjectured. Panel (b) of Figure 1.1 shows how finding a problem in QMA-complete∩BQP would make all problems in the drawn sets BQP, including NP-complete ones.

1.2 Ground state preparation in a digital

quan-tum computer

The Hamiltonian ground state problem is defined as follows. Given a Hamiltonian H on a n-qubit Hilbert space, prepare a n-qubit register in a state ρ0 that “approximates well” the Hamiltonian ground state |E0i. The figure of merit of the goal state can be defined differently depending on the application. If the full ground state structure is needed, we may be interested in getting arbitrary ground state fidelity hE0|ρ0|E0i. If we are rather interested in specific characteristics of the ground state, we may accept a state that approximates well enough the ground state expectation value of some class of operators (e.g. its energy and/or local operators).

As the Hamiltonian ground state problem is intrinsically quantum, 4

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1.2 Ground state preparation in a digital quantum computer 5

it seems natural to exploit a quantum algorithm to solve it. In specific cases (e.g. weakly correlated systems) the ground state form can be found classically, exactly or with good approximations (e.g. through mean field approaches or quantum Monte Carlo methods). In many cases, though, the exponential explosion of the Hilbert space needed to represent en-tanglement in a growing quantum system makes even the storage of the ground state classically unfeasible. As a n-qubit Hilbert space is spanned by N = 2n independent states, on a classical computer N −1 complex numbers have to be stored to represent an arbitrary state of the system, making the space requirements of any algorithm that aims to fully repre-sent any state O(2n), at least exponential. On a quantum computer, in-stead, n qubits are sufficient to store the system state. The cases in which the ground state displays a complex entanglement structure are thus the ones for which it’s most likely to get exponential speedup in preparing and characterizing the ground state by using quantum algorithms.

Ground state preparation is important for many applications. It can be used to initialize a register for other algorithms. Quantum Phase Estima-tion can be used to measure its energy, solving the lowest eigenvalue or ground state energy problem. The prepared ground state can be further characterized by measuring the expectation value of any observable of in-terest. The evolution of the prepared state under a different Hamiltonian can be simulated, and the response of some observable measured. As dig-ital quantum simulation is universal for local systems, results and appli-cations are not limited to qubits: one may map other system Hamiltonians to qubit Hamiltonians, extending our results to more general cases. For example, the Jordan-Wigner transform [21] allows to map Fermionic Fock spaces to qubit spaces, allowing to use quantum computation to tackle unsolved problems in theoretical chemistry [19, 22, 23], condensed matter physics and nuclear physics [24].

Finding the ground state of a generic k-local Hamiltonian has been proven to be a QMA-complete problem for k ≥ 2 [25]. Under more re-strictive conditions than k-locality on the Hamiltonian, although, it might be possible to find a polynomial-time algorithm that prepares the ground state with arbitrary fidelity without contradicting complexity theory con-jectures (see figure 1.1). For example, systems of n non-interacting par-ticles can be solved particle-by-particle, finally combining single-particle low energy states according to the relevant statistics to get the many-body ground state, in polynomial time. Recent research showed that all k-local Hamiltonians used in QMA-completeness proofs are critical [26], i.e. the energy separation between their ground state and first excited state van-ishes algebraically with increasing system size. An algorithm that promises

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6 Introduction

to find the ground state of some subclass of non-critical Hamiltonian would not threaten the current beliefs in complexity theory.

The aim of research on ground state preparation is twofold. From a complexity-theory point of view, we are interested in establishing broader classes of Hamiltonians for which the ground state can be obtained in polynomial time. From an applied quantum computing point of view, we are interested in finding methods and procedures that allow to ppare the ground state of limited-size Hamiltonians with the smallest re-source requirements, and that are robust against noise. This would allow to eventually implement these new ground state preparation algorithms in realistic Noisy Intermediate-Scale Quantum (NISQ) devices.

1.2.1 Existing techniques for ground state preparation

In this section we provide a brief summary of popular existing ground state preparation methods and algorithms. We focus on the principle be-hind each method, on their specific requirements (including classical pro-cessing, often part of these algorithms), their advantages and disadvan-tages.

Quantum Phase Estimation

Quantum Phase Estimation (QPE) [27] is one of the best known algorithms in quantum computation, and is often used as a key subroutine of other quantum algorithms. QPE directly performs a projective measurement of the phase of a unitary operator U, by entangling the system register with an ancillary register that stores the phase information and measuring the latter. By choosing the unitary operator U = e−iHt, the t-time evolu-tion operator based on the system Hamiltonian H, one would measure the phases by which the Hamiltonian eigenstates rotate under t-time evolu-tion. By performing QPE on a system register in the initial state|ψi and

post-selecting results with a minimum phase, one can project the system register on the ground state|φ0iwith probability|hφ0|ψi|2.

Post-selection is a requirement of QPE ground state preparation, which implies the need of an initial state |ψi with finite ground state overlap.

Such state may be obtained in some cases by classical approaches such as mean-field methods, or by other approximate preparation techniques such as the ones discussed in this thesis. QPE allows to distill results of approximate algorithm, turning a state with finite ground state overlap into an arbitrarily accurate ground state approximation. In its standard form, QPE requires deep circuits, noise is disruptive and it is expected 6

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1.2 Ground state preparation in a digital quantum computer 7

that QPE will need fault tolerant qubits to truly be effective [28]. Recently other QPE-like protocols such as Demon-Like Algorithmic Cooling [29] have been designed to try to mitigate the requirements of standard QPE.

Adiabatic state preparation

Methods in the branch of adiabatic quantum computing, like quantum annealing [9, 10], rely on a time-dependent Hamiltonian evolution and on the adiabatic following theorem. To prepare the ground state of a Hamilto-nian H, a quantum register is first deterministically prepared in the known ground state of an Hamiltonian H0. The state is then evolved according to a time-dependent Hamiltonian that changes slowly, starting in H0and morphing into the Hamiltonian H which ground state we desire to obtain. The adiabatic condition needs to be fulfilled throughout the computa-tion. To avoid Landau-Zener transitions, the Hamiltonian change needs to be slow at all times with respect to the instantaneous ground state gap [30]. This requires some degree of knowledge of the Hamiltonian spec-trum, and long evolution times. The definition of adiabatic conditions for specific classes of systems are object of current research [31]. The adia-batic evolution has to be discretized to be simulated on a digital quantum computer [32, 33], generally requiring a huge number of Trotter steps.

Variational algorithms with quantum subroutines

Variational quantum methods are among the most successful for near-term quantum computation [34–36]. These consist in a number of quantum-classical hybrid algorithms based on a parametrization of the state|ψ(θ)i =

U(θ)|0i(ansatz), that allows to approximate any state in the Hilbert space

by a applying a unitary operation U(θ)with a manageable number of

clas-sical parameters θ = (θ1, ..., θi) to a fiducial state |0i . Unitary prepara-tion of the ansatz and measurement of the Hamiltonian observable allow to use the quantum computer to perform the classically-hard subroutine needed to deduce the energy of the state given the parameters E(θ) =

hψ(θ)|H|ψ(θ)i. Variational Quantum Eigensolvers (VQE) are algorithms

where this subroutine is combined with classical optimization to minimize the energy E(θ). Once the optimal parameters are obtained, the

approx-imate ground state can be readily prepared by using the ansatz. Other notable variational algorithms that can approximate the ground state are the variational simulation of imaginary time evolution [18] and the Quan-tum Approximate Optimization Algorithm [36, 37].

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8 Introduction

The VQE is based on classical optimization in the high dimensional space of the parameters θ, which requires high classical computing re-sources and many calls to the quantum subroutine to prepare the state and measure E(θ). The estimation of energy E(θ) itself is a slow

proce-dure, as statistics over many measurements is required. The performance of VQS methods depends on the chosen ansatz, and on the structure of the ground state to be approximated.

Non-variational algorithms with classical feedback

In recent research, new classes of hybrid quantum-classical algorithms that can prepare Hamiltonian ground states are being developed. As an example, in the Quantum Imaginary Time Evolution (QITE) [38], the ground state component of any state is enhanced through a hybrid algorithm that simulates imaginary time evolution by progressively constructing a se-quence of parametric unitary operations. Parameters are adjusted deter-ministically based on information extracted from local tomography at all points of the construction. The resulting sequence of unitary operations effectively constitutes an ansatz optimized non-variationally.

As QITE requires state tomography to optimize each unitary operation in the sequence, part of the sequence will need to be repeated many times to prepare the state to be tomographed (this speed limitation is analogous to the one of the VQE). Any simulation of imaginary time evolution am-plifies the initial ground state component, so we need an initial state with a finite ground state overlap.

Analog simulation techniques

In the field of analog quantum simulators, various system-specific tech-niques to implement cooling to the ground state of a controlled quantum system exist. Their general description is beyond our scope, but in the context of this work it’s worth mentioning sympathetic cooling. This tech-nique consists in cooling down a controlled quantum system by letting it interact with a second one, that can be kept cold more easily. In the field of cold atomic gases, since its first implementation in 1997 [39], this tech-nique is routinely used to cool to a degenerate cold gas atomic species that cannot be easily cooled directly. Inspired by sympathetic cooling, in a re-cent exploratory work Raghunandan et al. studied analog simulation of dissipation mediated by a single qubit [40].

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1.3 Cooling in nature 9

1.3 Cooling in nature

In nature, physical systems are often coupled to a cold and large bath, and thus tend to their ground state. This dissipation of energy in the bath is represented by non-unitary open system dynamics, as coupling to a bath and then disregarding its state – in the formalism, “tracing out” the bath – is a non-unitary operation. As excitations move out of the system and into the cold bath, the system and the bath may entangle and the system entropy can increase. In the long time limit, however, the coldness of the bath will make the system tend to a low-energy fixed point. The final sys-tem state will be an approximation of the ground state with low entropy (i.e. low residual entanglement with the bath) under the following condi-tions:

• Coldness: the system spectrum has a large gap with respect to the bath thermal energy scale.

• Largeness: the bath has enough states not to return excitations to the system.

• Ergodicity: no symmetry is preventing the exchange of excitations and their diffusion in the bath.

1.3.1 The non-unitarity requirement

Unitary operations map states one-to-one, conserving information and en-tropy. To reach a specific pure state – the desired Hamiltonian ground state |φ0i – starting from an arbitrary and possibly mixed state, a non-unitary operation implementing a many-to-one mapping is required. If the start-ing state is mixed, the non-unitary algorithm will have the task to reduce the system entropy.

QPE implements non-unitarity through post-selection, discarding the state components orthogonal to the ground state. Adiabatic techniques don’t start from an arbitrary state: they require the initial preparation of a known pure state, and so they are able to get to the result through unitary evolution. Variational algorithms and QITE are based on classical updates to the ansatz rather than quantum evolution of a state: although these al-gorithms may be used to build a repeatable unitary routine that produces the target ground state from a specific pure state, non-unitarity in whole hybrid classical-quantum algorithm is provided by energy measurements, local tomography and preparation of the initial state.

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10 Introduction

1.3.2 Cooling from the Fermi golden rule

In physical systems that experience cooling by coupling to a cold bath, the system de-excitation is ensured by the conservation of energy. As long at the total energy is conserved, the system can loose energy to the bath and cool down, while the opposite process is statistically irrelevant because the bath is cold, i.e. devoid of excitations.

Typically, quantum systems cool down by coupling to a bath display-ing a continuum of states (electromagnetic field, phonon field), which is not excited at the energy scale of the system’s own excitations. We con-sider the bath to be initially in its vacuum state|0i_B. For |Eii_S the initial state of the system, with energy Ei, the instantaneous de-excitation behav-ior is captured by summing the Fermi Golden Rule over the possible final states of the bath:

dPi→f dt = 2π ¯h Z ∞ 0 de E_f, e V|E_i, 0i 2 δ Ei−Ef −e ρB(e), (1.1)

where ρBis the density of states of the bath, V is the perturbative coupling between the system and the bath, andE_f, e

=E_f

S|eiB is the final state with the system in the Ef energy eigenstate and the bath with a e energy excitation. The continuity of the bath spectrum grants meaning to the delta distribution that represents energy conservation. As seen in figure 1.2 (a), a system-bath coupling can cause an overall energy-conserving transition which lowers the system energy |E₁i_S → |E0i_S by creating an excitation in the bath|0i_B → |ei_B with energy e = E1−E0. As the bath is initially devoid of excitations, no energy-conserving transition that increases the system energy exist. In time-dependent perturbation theory – the frame-work in which the Fermi Golden Rule originates – energy conservation is derived as a stationary phase condition on the matrix element of the interaction-representation perturbing potential. The stationary phase re-quirement, lim t→∞ Z ∆t 0 e i t∆E dt 2 =2π t δ(∆E), (1.2)

involves a long-time limit. Furthermore, for the validity of the perturba-tive expansion it is required that

R∆t 0 V ∼ γ∆t 1, with γ parameter representing the scale of V.

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1.4 In this work 11

1.4 In this work

In this work we lay the bases for the design and construction of a class of approximate ground-state preparation algorithms based on an iterative simulation of cooling through a single-qubit fridge. We call these algo-rithms Quantum Digital Cooling, from here on QDC.

1.4.1 The quantum digital approach to cooling

With the aim of designing an efficient digital quantum algorithm for ground state preparation based on simulated cooling, we choose to implement the bath as a single two-level system. Simulating baths with a continuous spectrum on a digital quantum computer is impossible because of the dis-crete nature of computation, and any approximation would require large memory resources just to represent the bath. Conversely, a two-level bath requires only a one qubit overhead on the system simulation, its evolution may be simulated by an elementary rotation gate and its coupling to the system is easily implemented. To ensure its coldness, the qubit bath can be occasionally measured and reset to the low energy state. Under ap-proximate energy conservation conditions, resetting the bath to its ground

(a) _E tot e |E0i_S |E₁i_S |E₂i_S |E0iS |E1i_S |Eni_S ∆E ⊗ |0i_B ⊗ |ei_B (b) |E0i_S |E1iS |Eni_S |E0iS |E₁i_S |Eni_S eF ∆E ⊗ |0i_F ⊗ |1i_F

Figure 1.2: Energy level schemes for continuum (a) and discrete cooling (b). In

both cases, we consider the initial state is|E1i_S⊗ |0iand the energy-conserving

transition of the system to|E0iachieved by transferring an energy E1−E0in the

bath/fridge. In (a) the bath displays a continuum of possible excitations |ei_B,

while in (b) the fridge has a unique excited state at fridge energy eF, tuned to be

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12 Introduction

state prepares it to absorb more energy from the system: by analogy of this operation with cooling systems based on a thermodynamic cycle we will thus define the two-level bath fridge qubit. Figure 1.2 (b) shows the transfer of energy from a system to a single-qubit fridge: under the energy conser-vation condition, the fridge energy needs to be tuned to resonate with the transition energy to allow energy transfer.

The discrete nature of the bath requires a revision of the energy con-servation principle, as the product of the density of states with the Dirac delta distribution in equation (1.1) ceases to make sense. The long time limit in equation (1.2) needs to be dropped and time ∆t becomes a pa-rameter of the system: namely, the time between two instances of fridge measurement-and-reset. The energy delta function in the Fermi golden rule is substituted by Z ∆t 0 e i t∆E_dt 2 = 4 sin 2₍∆t ∆E 2 ) ∆E2 = sinc(∆t ∆E) 2 ∆t. (1.3)

For a finite time, the energy conservation will be satisfied up to a resolu-tion ∆E ∼ _∆t1. For a system with a discrete spectrum, this implies that transitions between eigenstates with energies that differ more than _∆t1 are strongly suppressed. Simulating cooling with a one-qubit bath on a pro-grammable quantum computer also gives us freedom in the choice of the interaction potential V. Having V and ∆t as parameters of our simula-tion allows us to go beyond the perturbative and continuous response of the Fermi golden rule, and a finite amount of energy and entropy can be moved out of the system in∆t time.

1.4.2 Contents by chapter

In chapter 2 we define, study and optimize the base element of QDC: the QDC cooling step. Its goal is to efficiently drive energy and entropy out of the system into the fridge qubit, which can then be reset discarding its state. To study the QDC step, we consider its application to a toy model consisting of a one-qubit system. We calculate the optimal form (V), strength (γ) and application time (∆t) of the coupling between the sys-tem and fridge. We describe how to implement the QDC step on a digital quantum computer with a discrete sequence of gates, and study the effect of discretization. We define two efficient implementations of the cooling step: one simpler and faster but more situational and prone to errors, and one more precise and tunable.

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1.4 In this work 13

In chapter 3 we develop the theory of QDC applied to many-qubit sys-tems. We define design principles for effective QDC protocols: well-defined algorithms based on iterating the cooling step while changing its param-eters. We explore ways to tune the fridge energy to resonate with dif-ferent transitions in the system spectrum. We describe caveats such as symmetries and localization and show how they can be mitigated. We de-sign and test two QDC protocols: LogSweep which is tunable but resource-expensive, and the bang-bang based protocol which is much faster and aims to produce coarser approximations of the ground state.

In chapter 4 we present the results of numerical simulations of QDC protocols on an interacting system model. We choose to use the transverse-field Ising spin 1/2 chain model, a simple and well known model of an interacting qubit Hamiltonian that can be parametrically tuned to different phases, across a quantum phase transition. We show the performace of both LogSweep and the bang-bang based protocol on the system in its different phases. We show the trajectories of energy, entropy and ground state fidelity of the system state during the application of the protocol, and we propose interpretations of the presented results. We characterize the cooling performance of both protocols on systems of sizes ranging from 1 to 10 spins.

In chapter 5 we present future perspectives of research on QDC and derived methods. We propose possible routes to improve and design new QDC protocols, and ideas for new quantum algorithms that integrate QDC principles with other known methods. We suggest that QDC may pave the way to a new class of non-unitary quantum algorithms based on ancilla-mediated non-unitary operations.

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Chapter

2

De-exciting a single transition

The goal of this chapter is to explore the possibility of de-exciting a single transition |E1iS → |E0iS in the spectrum of a system S, by simulating the open dynamics of the system coupled with a fridge qubit F, initialized in a specific state|0i_F. F and S subscripts will indicate the system and fridge subspaces throughout this thesis.

We want to design a procedure based on unitary operations on the sys-tem ⊗ fridge space that will maximize the |E0i component of the final system state ρ0_S, for as many initial states ρS as possible. The procedure consists in:

1. initializing the fridge in the state|0i_F,

2. applying a unitary USF, which will depend on the system Hamilto-nian, on the S⊗F space,

3. disregarding the fridge state by tracing out F.

This implements on the state of the system a process map E : ρS 7→ ρ_S0,

E [ρS] = TrFUSF|0iFρSFh0|USF† . The non-unitarity provided by points 1. and 3., and the dependence on the system Hamiltonian through the unitary evolution in 2. allow to realize an open-dynamics process that can distinguish system states by their energy.

ρS

USF(t)

ρ0_S = E [ρS]

//

|0i dump

We call this procedure the cooling step. It will serve as the building block for our Quantum Digital Cooling (QDC) protocols.

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16 De-exciting a single transition

2.1 Minimal model

One may represent the single transition we want to cool |E1iS → |E0iS in a one qubit model system with Hamiltonian HS = 1₂ESZ. Here, ES = E1−E0, and the two states in the transition are represented by |1i_S and |0i_S, eigenstates of the third Pauli matrixZ. The fridge qubit Hamiltonian is defined as HF = 1₂eZ. Throughout this thesis,{X,Y,Z }will represent Pauli matrices.

For an efficient energy exchange, the fridge energy splitting is tuned to be resonant with the system transition e = ES. To couple the fridge and the system efficiently, we choose a coupling potential that anticommutes with both Hamiltonians: V = 1₂γXSXF, where γ is the coupling strength.

The resulting total Hamiltonian is block diagonal:

H = HS+HF+V comp. basis −−−−−−→     e γ₂ γ 2 γ 2 γ 2 −e     , (2.1)

separating two subspaces spanned by{|00i_SF,|11i_SF}and{|10i_SF,|01i_SF}. As the fridge is initially set in |0i_F, the S⊗F transitions that can happen are|10i_SF→ |01i_SFand|00i_SF→ |11i_SF. We want the first transition to be optimally realized to de-excite the system, while preventing the second. We call the energy non-conserving process|00i_SF → |11i_SF reheating, and we shall try to avoid it. The Hamiltonian generates an evolution operator with the same block diagonal structure,

e−i H t →    cos(Ω t) + e Ωi sin(Ω t) 2γΩi sin(Ω t) cos γ 2t i sin γ 2t i sin γ 2t cos γ 2t γ

2Ωi sin(Ω t) cos(Ω t) + Ωe i sin(Ω t)

  , (2.2) whereΩ= q γ2 4 +e2.

By imposing |0i_F initial state of the fridge and tracing out the fridge after evolution, we obtain the process map

E [ρS] =TrF h

e−i H t|0i_FρSFh0|ei H t i

. (2.3)

By tuning the parameters of the process – the evolution time t and the coupling strength γ – we aim for our process map to lead any initial pure or mixed state to a low energy and entropy state with a large de-excited component h0|E

ρS

|0i. 16

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A quantum process can be represented by a Pauli Transfer Matrix (PTM) [41], a matrix that maps a decomposition of the initial state in the Pauli strings basis to the same decomposition of the final state. We here use a generalized definition of PTM

(E_PTM)_ij =TrhPiE [Pj] i

(2.4) where the decomposition basis is picked to be Pi ∈ {|0ih0|,X,Y,|1ih1|}. This basis allows to easily read off the amplitude of transitions|0i → |1i and|1i → |0i. The PTM representing E in this basis is

  

cos2₍_tΩ_{) +}e2sin2 (tΩ)

Ω2 0 0 γ2sin2 (tΩ)4Ω2

0 2 costγ₂cos(tΩ) +γsin

_tγ 2 sin(tΩ) Ω 2e costγ₂sin(tΩ) Ω 0 0 −2e cos _tγ 2 sin(tΩ) Ω 2 cos _tγ 2 cos(tΩ) −γsin _tγ 2 sin(tΩ) Ω 0 sin2_tγ 2 0 0 cos2_tγ 2    . (2.5)

2.1.1 The two implementations of the cooling step

The inner block of the evolution operator (2.2) describes the Rabi oscilla-tions corresponding to the cooling transition|10i_SF ↔ |01i_SF. The crest of the Rabi oscillation is found at

γ t =π. (2.6)

Fixing this makes the evolution swap the states in the inner block, maxi-mizing the probability of transitioning from|1i_S to|0i_S:

h0|E |1ih1| |0i = sin2γ 2 t →1. (2.7)

and leading to the process map represented by the PTM

    1− γ2 4Ω2 sin2(tΩ) 0 0 γ2 4Ω2 sin2(tΩ) 0 γ Ωsin(tΩ) 0 0 0 0 −γ Ωsin(tΩ) 0 1 0 0 0     . (2.8)

Similarly, the outer block of the evolution operator (2.2) describes the energy nonconserving transition|00i_SF ↔ |11i_SFthat generates reheating of the system. The probability of transitioning from|0i_Sto|1i_S is

h1|E |0ih0|

|1i = γ

2_sin2_(t_Ω₎

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We want this probability to be small. There are two approaches we can take to minimize this reheating, which in turn define two possible im-plementations of the cooling step. We will investigate and extend these throughout this thesis.

Our first approach to keep (2.9) small, is to make γ/Ω small by choos-ing γ e. This weak coupling approach reduces the amplitude of the Rabi

oscillations between the states|00i_SF ↔ |11i_SF, thus reducing the reheat-ing probability. To set the parameters for the coolreheat-ing step, we combine the weak coupling condition with (2.6), obtaining

e =ES, γ= π

t e, te

−1

. (2.10)

The long time and weak coupling allows us to realize the resonant tran-sition|10i_SF → |01i_SFwhile hindering the non-resonant |00i_SF → |11i_SF, implementing a form of energy quasi-conservation. As suggested in sec-tion 1.4.1, energy quasi-conservasec-tion holds to a linewidth δ ∼ t−1: the freedom we have in choosing t and γ can be used to define such linewidth. The PTM that represents the cooling step under the conditions of equa-tion (2.10), to first order in γ2

e2 is     1− γ2 4e2 sin2(te) 0 0 γ2 4e2 sin2(te) 0 γ e sin(te) 0 0 0 0 −γ esin(te) 0 1 0 0 0     +O(γ 4 e4). (2.11)

The other approach to minimize (2.9) consists in fine-tuning the evolu-tion time to suppress the off-resonant reheating process, tΩ = π.

Com-bined with (2.6), this fixes all parameters of our cooling step:

e =ES, γ= √2 3e, t= √ 3 π 2 e −1_. _(2.12)

At the cost of being farther away from energy conservation and leaving us without any free parameter, the strong coupling regime allows the imple-mentation of a cooling step in a very short evolution time. The PTM of the cooling step under these conditions is

    0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1     (2.13) 18

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2.1.2 Scaling with transition energy

Both implementations of the cooling step require t to scale with E−_S1, thus requiring longer times for cooling smaller transitions. We suggest this is indicative of a fundamental principle: to distinguish two states by the phase they acquire through Hamiltonian evolution, we need time O(∆E−1) to learn the phase difference up to measurable resolution and determine which state is higher in energy. This knowledge is fundamental for any cooling process that aims to reach the ground state independently of the structure of the Hamiltonian eigenstates. The only meaningful element that can be used to distinguish states in this case is their energy.

This is the first hint that small energy separation will be a problem for ground state preparation through cooling. We know that all known Hamiltonians tha generate QMA-complete problems are critical [26]. In critical Hamiltonians for which the gap scales algebraically with the size of the system n, it should still be possible to distinguish the lowest states in time poly(n), so this principle is not sufficient to claim that critical Hamil-tonians cannot be cooled to the ground state in a polynomial time. A principle that connects QMA-hardness the difficulty to cool down systems with critical spectra would give a deep insight in the nature of quantum complexity.

2.1.3 Discretization

We just described the continuous evolution of a one qubit system coupled to one qubit fridge and how this could be used to de-excite a transition of the system by transferring energy and entropy to the fridge. To implement this on a digital quantum computer we need to decompose the continuous evolution in an algorithm, built as a finite sequence of available gates.

We can approximate the Hamiltonian evolution of the system e−iHst_for

an arbitrary time t as a composite gate sequence. The evolution of a system with a k-local Hamiltonian can be approximated with arbitrary precision by a Trotter expansion [42, 43], as a sequence of k or less-qubit rotations of polynomial length in the number of qubits. For k constant under system scaling (local interactions), this allows to realize e−iHst_{, i.e. Hamiltonian}

simulation, in polynomial time. Extended and optimized Hamiltonian simulation methods have since been developed [44, 45]. In this thesis the error deriving from this approximation is not considered it, and the uni-tary e−iHst_{is considered as one gate on the register representing the system}

state. We also assume to have full control on the fridge qubit: we can real-ize any one-qubit rotation on it and any two-qubit rotation involving the

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fridge and one of the system’s qubits.

For the one-qubit system, the Hamiltonian evolution of the system e−i Hst _{is the one qubit rotation on the Z axis of phase E}

st. To simulate the fridge’s own evolution we will use e−i HFt_{. The coupling evolution}

e−i V tconsists of a two-qubit rotation that can easily be implemented on a digital quantum computer.

To discretize the cooling step simulation, we use a variation of the the Trotter approximation: e−i(HS+HF+HSF)t _∼_e−i HSF _M+1t h e−i HS Mt e−i HFMt e−i HSF _M+1t iM , (2.14) which is exact in the large N limit. The choice of having e−i HSFMt at the

beginning and end of the sequence is due to the fact that e−i HF _M−1t _would

have no effect at the beginning of the sequence (it would act on|0i_F) nor at the end (the final state of the fridge is discarded).

The choice of the number of Trotter steps M is of great importance: it needs to be large enough to make the cooling step work as wanted, but as low as possible to reduce the computational resources needed to imple-ment it. On the subspace relative to the cooling transition|10i_SF ↔ |01i_SF, all matrix elements of HS and HF are equal to 0. If we restrict to this subspace, all terms in the Trotter expansion commute and M = 1 step is enough implement the cooling transition exactly. The choice of M de-pends on the deviation from the continuous limit of the probability of the reheating transition |00i_SF → |11i_SF. Observing the effect of M on the reheating transition, in Fig. 2.1, we notice that the reheating probability becomes critically large at the M-th oscillation, where cooling and reheat-ing become equiprobable. The reheatreheat-ing probability remains small for the first m M oscillations. The number of reheating oscillation crests before the working point of the cooling step, γ t=π, is

m = tΩ π = tpγ2+e2 π = s 1+ e 2 γ2. (2.15)

To implement safely the cooling step within the weak coupling regime it will suffice to choose a number of Trotter steps Mm. If we are dealing with a system with more than one transition coupled to the fridge, extra care needs to be taken: larger-energy transitions might be off-resonantly reheated if an insufficient number of steps M is selected. In practical ap-plications we find that, even in these cases, a sufficient number of steps to 20

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2.1 Minimal model 21 0 2 3 t M = 1 M = 2 M = 3 M = 4 M = 5 M = 6 M = 7 cont. M |0 |1 |1 |0 0 2 3 t M = 5 M = 10 M = 15 cont. M |0 |1 |1 |0

Figure 2.1: Probability of cooling (dashed) and reheating (solid) transitions vs coupling time t, for varying number of Trotter steps M and for continuous evo-lution (in black). Left and right panels only differ for the choices of M. The

cou-pling strength is fixed γ = 0.2 e and the dotted line indicates the working point

γ t = π. The curves for different M values are offset for readability, the dashed

curves (cooling probability) oscillate always from 0 to 1. The dots mark the M-th reheating Rabi oscillation crest.

suppress reheating is M ≥2 s 1+ (e/2+Emax/2) 2 γ2 , (2.16)

where Emax is the largest of the energies of transitions coupled to the fridge.

In the strong coupling regime a bang-bang approach to discretization can be taken: an application of the local evolution e−i HSt_e−i HFt

sand-wiched between two applications of the coupling evolution e−i HSFt

(equiv-alent to equation (2.14) with M=2) is sufficient as long as the parameters

γ and t are re-adjusted. This gate sequence, with the condition γ t = 2π,

implements Ramsey interferometry simultaneously on the two transitions |10i_SF ↔ |01i_SF and |00i_SF ↔ |11i_SF. Separately for each transition, the Hamiltonian evolution is sandwiched between two π/2 pulses. This pro-duces a flip on the resonant cooling transition (for which the splitting is zero), and a identity operation on the off-resonant reheating transition if 2ESt = π (so that the local evolution implements a phase flip). The

pa-rameters for the strong-coupling bang-bang cooling step should thus be chosen as

e =ES, γ=2 e , t = π

2 e

−1_. _(2.17)

The effect of the continuous and bang-bang strong-coupling cooling steps are shown in Figure 2.2: the optimal Rabi oscillations in the continuous case (left panel, black curves) are reproduced by Ramsey fringes (blue curves) under adjusted parameters (right panel).

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0 2 3 t

M = 1

cont.

M

Parameters for continuous cooling step ( / = 4/3)

|0 |1 |1 |0 0 2 3 t M = 1 cont. M

Parameters optimized for Ramsey-based step ( / = 2)

|0 |1 |1 |0

Figure 2.2: Probability of cooling (dashed) and reheating (solid) transitions vs

coupling time t, for bang-bang approach M = 1 (blue) and for continuous

evo-lution (black). The dotted line indicates the working point γ t = π. The curves

are offset for readability, the dashed curves (cooling probability) oscillate always from 0 to 1. Left: the parameters that minimize reheating for continuous evo-lution, as in equation (2.12), are selected. Right: the parameters that implement anti-resonant Ramsey interferometry on reheating, as in equation (2.17), are se-lected.

2.2 Detuning

In general, we do not know the exact energies of the transitions in the spec-tra of the systems which we are interested to cool. In some case we might be able to approximately predict these energies, else we might just try to cool transitions at many energy values, picked systematically in some spectrum. Theory and protocols for choosing energies will be developed in the next chapter. In any case, we need to consider what happens if

e 6= ES(i.e. we pick a fridge energy e which is not resonant with a system transition to which the fridge is coupled). We may define the detuning between the fridge and the transition energy∆= ES−e.

The effect of detuning in the weak coupling case is shown in the right panel of figure 2.3. Detuning reduces the efficiency of cooling, as it moves the cooling transition|10i_SF↔ |01i_SFout of resonance. The cooling prob-ability is large in a region of small detuning scaling as t−1, i.e. cooling becomes ineffective if∆ & t−1. However, the Reheating probability stays small even upon detuning. Positive detuning will shift the off-resonant reheating transition even more off-resonance, while upon negative detun-ing, naturally limited to∆ = −e, the reheating transition energy 2e−∆ is

always kept at least e far from resonance. At maximum negative detuning ∆ = −e the cooling and reheating probabilities are equal, as the system

transition has zero splitting, and |0i_S and |1i_S can’t be distinguished by their energy. Because of the smallness of reheating probability we can af-ford to repeat the cooling step at different fridge energies, increasing the 22

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2.2 Detuning 23 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Detuning / 0.0 0.2 0.4 0.6 0.8 1.0 transition probability |1 |0 |0 |1 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Detuning / 0.0 0.2 0.4 0.6 0.8 1.0 transition probability t = 5 1_{= 0.63} t = 10 1_{= 0.31} t = 20 1_{= 0.16} |1 |0 |0 |1

Figure 2.3:Effect of detuning between fridge and transition to be cooled on cool-ing probability (dashed lines) and reheatcool-ing probability (solid lines). Left: bang-bang cooling step, with parameters (2.17). Right: continuous weak-coupling cool-ing step, with parameters (2.10).

chance of hitting the resonant energy and instantly cooling the system, without worrying that the reheating process would spoil the effort: this is the principle at the heart of log-sweep protocol that we will introduce in section 3.1.

In figure 2.4 we show that, even in the presence of detuning, discretiza-tion as in equadiscretiza-tion (2.14) preserves the smallness of reheating probability. However, the number of steps M must be chosen taking into account that the maximum system energy depends on the maximum possible detuning Emax =e+∆max. To perform cooling in a finite amount of steps, we surely need an upper bound on the energy of transitions in the system spectrum that V couples to the fridge.

In the bang-bang strong-coupling implementation of the cooling step, the minimization of the reheating probability is based on fine tuning of the system parameters. The large coupling strength and the short

cou-1.0 0.5 0.0 0.5 1.0 1.5 2.0 Detuning / 0.00 0.02 0.04 0.06 0.08 0.10 0.12 re he at ing |0 |1 p ro ba bil ity t = 5 1_{= 0.63} t = 10 1_{= 0.31} t = 20 1_{= 0.16} continuous trotterized

Figure 2.4: Comparison between reheating probabilities in presence of detun-ing for continuous and trotterized implementations of the weak coupldetun-ing cooldetun-ing step. Parameters coincide with Fig. 2.3 (right panel), for the discretized version

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pling time prevent us from using energy conservation-like arguments to prevent reheating. As shown in the left panel of Figure 2.3 detuning in-creases reheating probability as much as it dein-creases cooling probability. However, the large linewidth associated with the violation of energy con-servation allows the cooling to be efficient at small detuning in a wider energy range around e than the weak-coupling cooling step. For this rea-son, the bang-bang approach is extremely efficient if we want to perform rapid cooling on a transition with an energy we approximately know.

2.3 Unknown eigenstates

It might happen that we know – approximately or even precisely – the energy of a transition, but we have partial or no information about the eigenstates |E0i and |E1i. In the model one-qubit system, random orien-tation of Hamiltonian eigenstates is represented by a randomly directed Pauli operator:

Hs =Es1

2h·σ =Es 1

2(cos(θ) Z +sin(θ)cos(φ) X +sin(θ)sin(φ) Y ) (2.18) where h is a random vector on the unit sphere, with θ and φ the associated polar and azimuthal angles.

As the system Hamiltonian is not proportional toZanymore, the system-fridge coupling should be chosen differently

V = γ

2 XFVS, (2.19)

where, in the one-qubit model case, the system-side coupling potential VS = v·σ is a Pauli operator of which we can choose the direction v. As we have complete control on the fridge, and its Hamiltonian is always HF ∝ Z, there is no need to change the F-subspace part of V. At the beginning of this chapter we choose VS = XS so that it would anticom-mute with HS(satisfying the condition v ⊥h), making sure that it would mix efficiently the Hamiltonian eigenstates. As we now don’t know HS, whichever system-side coupling potential VS we choose, there is a possi-bility it will commute with HS: in this case the system eigenstates won’t be mixed and no cooling can be performed. More generally, v (VS) will neither be parallel (commuting) or perpendicular (anticommuting) with h (HS), this will make the cooling less efficient.

The step can be repeated to get closer to the ground state, if the re-heating probability is kept low (e.g. by applying the weak-coupling cool-ing step) and the coolcool-ing probability is finite. The left panel of Figure 2.5 24

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2.3 Unknown eigenstates 25

shows that this approach results in a modest increase of the average cool-ing probability.

Another approach to increase the average single-step efficiency would be increasing γ: the effective coupling strength will be rescaled as a func-tion on the projecfunc-tion of v on the plane perpendicular to h, so on average the effective coupling strength will be smaller than intended. This can be partially compensated by increasing γ, as seen in Figure 2.5 right panel. This produces a modest increase in average cooling, but also in the re-heating probability, for which reason it doesn’t combine well with the step repetition.

As long as a fixed potential VS is choosen, the worst-case Hamiltonian (for h k v) and the ones with a close enough h will be cooled very ineffi-ciently. However, as random point on a sphere is more likely to be close to the equator (h almost perpendicular v) than to the poles (h almost paral-lel to v), the average coupling efficiency is higher than one could naively expect.

0 /2

angle between v (VS) and h (HS)

0.0 0.2 0.4 0.6 0.8 1.0 transition probability t = 5, = 0.63 1 step 2 steps 3 steps |1 |0 |0 |1 averages 0 /2

angle between v (VS) and h (HS)

0.0 0.2 0.4 0.6 0.8 1.0 transition probability t = 5 t = t = 4/3 |1 |0 |0 |1 averages

Figure 2.5: Cooling (dashed lines) and reheating (solid lines) transition proba-bilities for varying angle θ between v and h. The gray vertical lines indicate equal θ-sampling probability regions, for v and h randomly distributed on the unit sphere. Dotted lines indicate averages of transition probabilities weighed

by sampling probability. The coupling γ e is here choosen rather large for

the sake of plot clarity: a lower value would reduce reheating. Left: the cooling step is repeated up to three times. Right: a slight increase in gamma can also increase the average cooling probability. The case that maximizes the average

cooling probability γt= √4/3 π is shown.

2.3.1 Beating the symmetries by changing coupling

Until now we kept the system-side coupling potential VS fixed. Varying it can solve the symmetry-related problems that didn’t allow us to cool the system in the case v k h. In the weak-coupling regime we can apply

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the cooling step repeatedly without fearing reheating. As we have com-plete control over the step parameters, among which VSis, we can choose to vary VS at each iteration. We can choose, for example, to alternate VS among three orthogonal Pauli operators{X_S,Y_S,Z_S}: this way, if H_S com-mutes with one of these, it anticomcom-mutes with the two orthogonal ones. In figure 2.6 we show the result of applying in sequence cooling steps with this choice of VSto all possible system Hamiltonians of form (2.18), repre-sented on the surface of a sphere. Already after two steps with orthogonal VSchoices, no point on the sphere is left uncooled. The iteration of cooling steps changing coupling potentials can be used also to cool multiple-qubit systems that might display localization (see section 3.3).

X Y Z avg P(|0 |1 ) = 87% ± 22% min P(|0 |1 ) = 0% 0.20 0.40 0.60 0.80 1.00 X Y Z avg P(|0 |1 ) = 98% ± 1% min P(|0 |1 ) = 93% 0.94 0.96 0.97 0.99 X Y Z avg P(|0 |1 ) = 98% ± 1% min P(|0 |1 ) = 97% 0.97 0.97 0.98 0.98 0.99

Figure 2.6: Cooling of transition probability after applying, in sequence, three

cooling steps with VS1 = XS, VS2 = YS, VS3 = ZS (left: only first step applied,

center: first two steps applied, right: all three steps applied), for all possible val-ues of the unit vector h represented on the surface of a sphere. The red-green-blue dashed lines represent unit vectors in the ˆx- ˆy- ˆz directions respectively.

In this chapter, we have developed two implementations of a quantum digital cooling step, and shown that they can be used to coo an isolated transition efficiently, without full prior knowledge of the eigenstates or the transition energy. We have shown that we can choose step parameters to keep reheating low. Finally, by changing the fridge energy and the cou-pling potential, we have overcome symmetries and the energy mismatch that can hinder the cooling process. In the following chapters we shall ex-tend the cooling step ideas to define protocols that allow simulated cooling of larger systems.

26

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Chapter

3

The theory of Quantum Digital

Cooling

In the previous chapter, we introduced the idea of Quantum Digital Cool-ing and showed how it can be applied to cool a two-level system. We argued this can be treated as a toy model of a single transition in a larger system. However, when embedding this two-level transition model in a larger Hilbert space, some complications arise.

• To get from an arbitrary state to the ground state of a many-level system, several transitions need to be sequentially de-excited. These transitions can have different energies and may require different po-tentials V to be coupled to the fridge.

• The fridge qubit can be coupled to many transition by the coupling potential V, causing possible interference effects which complicate the understanding and treatment of the problem. However, this can be seen as an advantage as a single cooling step could be used to de-excite one of many possible transitions, reducing the number of steps needed. This is specially true for the cooling of mixed states, which components may need different transitions to be de-excited, and don’t interfere with each other.

• The space of choices for the system-side coupling potential VSis also much wider in the many-level system case. For simplicity of imple-mentation on a quantum computer, we restrict to coupling the fridge to one of the system qubits at a time.

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28 The theory of Quantum Digital Cooling

of QDC protocols for multi-qubit systems, discuss ideas and theoretical arguments behind their formulation and their expected validity.

3.1 Sweeping the fridge energy

As discussed in the previous chapter, both implementations of the QDC step are efficient in cooling a transition if we (approximately) know its energy. In this case we can tune the fridge energy e to match the transition energy within a linewidth δ ∼ 1/t. To find the ground state of a many-qubit system, we expect to have to match many transitions with different and possibly unknown energies.

A possible strategy to do this involves iterating cooling steps while sweeping e between some minimum and maximum energy Eminand Emax. These define the range in which we expect energies of all transitions to which the fridge can be coupled by a local VSare. In this approach, only a fraction of the cooling steps are expected to match any relevant transition and cool the system efficiently. Many cooling steps will be off resonance with all transitions, and might only cause reheating. The weak coupling cooling step (2.10) minimizes reheating of off-resonant transitions, so it will be the choice in protocols based on fridge energy sweeping.

3.1.1 Design principles

To design a QDC protocol based on iteration of the weak coupling step (2.10) with different fridge energies, we need to choose for each cooling step i∈ {1, ..., Nsteps}the fridge energy ei, the coupling time ti, strength γi and form VSi, and the number of trotter steps Mi.

We consider a simple model system to cool down, consisting in n non-interacting qubits, each with a random energy E in the range(Emin, Emax) and Hamiltonian proportional toZ(we could extend this to random Hamil-tonian orientation following the considerations in section 2.3.1). This way, we can cool one qubit at a time choosing the coupling potential VSij = X_Sj = 1_S1⊗1_S2⊗...⊗ X_Sj ⊗...⊗1_Sn as a one qubit operator acting on the j-th system qubit, and repeating the procedure for each j. In this case VSij is constant with respect to the index i, representing the iteration of the QDC step needed to cool a single qubit. This simple model system allows to study the problem of multiple transition energies without need to treat the case-specific interference problems. We also suggest that re-sults obtained on this model could be extended to other non-interacting or weakly interacting models.

28

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3.1 Sweeping the fridge energy 29

as discussed in section 2.2 and shown in figure 2.3, a single cooling step will cool down a qubit efficiently if its energy E is close to the fridge energy e, within linewidth δ scaling as ∼ 1/t. To have a chance to cool the qubit efficiently for all possible energies E, we want to pick a set of

ei and relative δi that covers the whole range (Emin, Emax), meaning that ∀E ∈ (E_min, Emax),∃i : ei−δi ≤ E≤ei+δi.

Picking a smaller linewidth δ (i.e. increasing the step time t and de-creasing the coupling strength γ) will require more cooling steps to cover the energy range, but it will also decrease the reheating probability for all possible E values. This, in turn, will allow for more off-resonant steps to be performed without causing reheating, a necessary condition for a protocol based on iteration of the cooling step. We want to keep the reheating prob-ability low specially towards the end of the process, while we can afford larger reheating at the beginning, as we can later cool down what has been reheated. We thus aim to progressively reduce the linewidth δ during the cooling step iteration.

As seen in figure 2.3, the qubits with lower energies are the ones that suffer the most from reheating. Thus, we want to cool them at later iter-ations to keep the probability of reheating them after being cooled down low. For this reason, when iterating the cooling step we want to sweep the fridge energy e from larger to smaller values.

3.1.2 The LogSweep QDC protocol

We here define a choice of cooling step parameters based on the principles above, meant to cool a qubit (or, by repetition, multiple non-interacting qubits) with random energy within a range, to arbitrary ground state over-lap. The input variables needed to set up the protocol are the energy range limits Emin, Emax, the number of cooling steps Nsteps, and the proportion-ality factor α ≡t δ which defines the concept of linewidth.

We pick{ei}as Nstepslogarithmically spaced points between e1 =Emax and eNsteps = Emin:

ei =E i−1 Nsteps−1 min E 1− i−1 Nsteps−1 max , (3.1)

and{δi}to satisfy the condition

ei+1+δi+1=ei−δi =⇒ δi =ei 1−

2

1+ (Emin/Emax)1/(1−Nsteps) !

. (3.2) The choice of{ei}and{δi}is schematically represented at the top of plots in figure 3.1. The coupling time and strength are then defined as ti =α/δi

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30 The theory of Quantum Digital Cooling

and γi =π/ti. The number of Trotter steps per cooling step Miis defined according to equation (2.16): Mi =2 s 1+(ei/2+Emax/2) 2 δ2_i π2 . (3.3)

The time required to implement the protocol on is then∝ ∑_iMiper system qubit.

In figure 3.1 we show how each single step i and the whole protocol (it-erating over i) act on qubits with different energy splittings ES. To realize cooling on the whole energy spectrum, the protocol combines the effec-tiveness of each step in cooling different sections of the spectrum. The cooling efficiency of the protocol decays fast outside of the energy range (Emin, Emax). Increasing the number of cooling steps Nsteps decreases the reheating probability for each step, and increases the cooling porbability of the protocol on the whole energy interval.

0 2 4 6 8 10 12 14 Transition energy ES 0.0 0.2 0.4 0.6 0.8 1.0 transition probability

total protocol trotter step count: 30

|1 |0 |0 |1 0 2 4 6 8 10 12 14 Transition energy ES 0.0 0.2 0.4 0.6 0.8 1.0 transition probability

total protocol trotter step count: 124

|1 |0 |0 |1

Figure 3.1: LogSweep protocol for Emin = 1, Emax = 10, α = 1 and two choices

of Nsteps = 5 (left), Nsteps = 10 (right). In the top part of the plot, the colored

bars represent the choices of ei±δi, with the blue bar representing the first step

(e1 = Emax). The colored lines represent cooling (dashed) and reheating (solid)

probabilities for varying qubit energy ES, when only applying the i-th cooling

step with parameters ei and δi represented by the same-colored bar. The dashed

black line represents the cooling probability when applying the whole protocol, i.e. iterating the cooling step over i.

3.1.3 Scaling of the LogSweep protocol

The final ground state overlap we can achieve for a n noninteracting qubit system cooled down by the LogSweep protocol depends on the number of cooling steps performed Nsteps, and on the number of qubits n. In turn, the total number of Trotter steps Mtot, which are proportional to the gate count 30