Variational Quantum Eigensolver for the XX-model

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Variational Quantum Eigensolver for the XX-model

Paco Bontenbal

May 30, 2021

Abstract

Variational Quantum Eigensolvers (VQEs) are quantum-classical hybrid algorithms that look to be very promising at estimating ground state energies of quantum mechanical systems, despite the shortcomings of the Near-term Intermediate Scale Quantum (NISQ) devices, like gate noise, and decoherence. We propose guidelines for using the VQE on a Quantum Processing Unit (QPU) to solve a specific problem. We present the accuracy of the VQE’s ground state estimations for spin-1/2 chains according to the XX-model on a Quantum Virtual Machine (QVM), with and without a noise model present. Furthermore, we show the success rate of Bell Singlet preparation on Rigett’s Aspen-9 QPU.

Student number 11230967

Daily Supervisor Joris Kattem¨olle

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Populair wetenschappelijke samenvatting

In de jaren 40 waren computers nog zo zwaar als een stoomlocomotief, vulden ze hele gebouwen en dat om genoeg rekenkracht te hebben voor slechts 5000 optelsommen per seconde. Nu, 80 jaar later, verlaat niemand zijn of haar huis zonder ´e´en of meer apparaten met een ongelooflijke rekenkracht. Deze toegenomen computerkracht in steeds kleinere computers is onder meer te danken aan de krimp van de transistor, de fysieke representatie van de 0’en en 1’en, of bits, waarmee computers informatie kunnen opslaan en verwerken. Deze krimp is echter tegen een limiet aangelopen, nu de transistors zijn gekrompen richting het formaat van atomen. En dan krijg je met quantum-mechanica te maken, waar de wetten van de natuur niet hetzelfde zijn als de natuurwetten die wij ervaren. In de afgelopen jaren zijn eigenschappen van juist deze quantum-mechanica gebruikt voor het ontwikkelen van quantum computers, bestaande uit qubits in plaats van bits. Deze qubits hebben de eigenschap dat ze niet alleen 0 of 1 kunnen zijn, maar zich in een superpositie van beide kunnen bevinden. Deze andere manier van informatie opslaan en verwerken zorgt ervoor dat quantum computers bepaalde berekeningen uit kunnen voeren die zelfs voor klassieke supercomputers niet binnen redelijke tijd gedaan kunnen worden. Maar, om dat te bereiken moeten quantum computers meer qubits hebben dan nu het geval is. We leven in het Noisy Intermediate-Scale Quantum (NISQ) tijdperk van quantum computing, waar het aantal qubits beperkt is en ruis resultaten erg kunnen be¨ınvloeden. Een algoritme dat nuttige resultaten kan leveren ondanks deze beperkingen is de Variational Quantum Eigensolver (VQE). In dit verslag komt aan bod wat de tekortkomingen van het NISQ tijdperk zijn, waarom een VQE toch nuttige resultaten produceert en hoe een VQE werkt aan de hand van het gebruik van een VQE voor een specifiek optimalisatie probleem. Hieraan voorafgaand zal een introductie worden gegeven over quantum computing en andere concepten die nodig zijn voor het begrijpen van een VQE. Het verslag zal worden afgesloten met een blik op de toekomst met betrekking tot quantum computing en de rol van VQEs daarin.

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1 Introduction

Quantum computers approach solving problems in a new way compared to classical comput-ers. Classical computers contain micro processors, that nowadays contain billions of transis-tors. These can either be turned on or off to represent its two states, 0 and 1. This allows them to store data using binary digits, or bits. Quantum computers however, use qubits to process information. These qubits are able to be in a superposition of 0 and 1, not being either one. Upon measurement the qubit’s superposition collapses, with a probability of it being in the 0 state and a probability to be in the 1 state. Quantum computers are potentially able to process a lot more data possibilities than classical computers, because of these superpositions, together with other quantum mechanical phenomena like parallelism and entanglement [1]. Now, this does not mean that quantum computers will replace classical computers. They will, most likely work alongside them, solving tasks too hard for classical computers.

Some applications of quantum computers have theorised to be; quantum chemistry [2], mod-elling of physical systems at quantum mechanical levels [1], but also the improvement of algorithms useful for the realisation of quantum neural networks [3], just to name a few. The speculation of the potential of quantum computing has inspired a lot of companies to explore quantum computing. Google, IBM, IonQ, Alibaba, Honeywell, Quantum Inspire and Rigetti and more provide quantum computing services. However, today’s quantum computers are not yet capable of useful quantum supremacy [4]. Still, they can give interesting results with the use of smart algorithms. One of these algorithms is the VQE [5, 6]. In this paper, the VQE will be discussed in depth and used for finding the ground state energies of spin-1/2 chains [7] according to the XX-model. Also, both the concept of Noisy Intermediate-Scale Quantum (NISQ) and the VQEs role within the NISQ era will be discussed.

2 Introduction to Quantum Computing

Just like the transistor physically realises the bit, the qubit also has a physical realisation. It can be realised multiple ways. Some examples are: photon polarisation, trapped ion energy levels or persistent current in superconducting circuits [1]. In this paper all of the QPUs discussed have superconducting qubits. Like classical computers, quantum computers use operations on one or multiple qubits to change the information stored in them. However, these operations and their representations are different.

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2.1 Single qubits and single qubit gates

The information about a single qubit is represented by its statevector that represents all the states the qubit can be in and the probabilities for those states. A statevector is usually represented with the Dirac notation of a vector in quantum mechanics, and for the states 0 and 1 look like

|0i = " 1 0 # , |1i = " 0 1 # . (1)

The qubit is the simplest form of a quantum system and its state, if normalised, can be expressed as

|qi = α |0i + β |1i , (2) where |qi is the statevector of a qubit, and |α|2 and |β|2 are the probabilities to obtain |0i or |1i, respectively. We can express any qubit this way because |0i and |1i form an orthonormal basis. Normalised here means that the value of α and β can be anything as long as |α|2+ |β|2 = 1 and α, β ∈ C. This restriction, together with the fact that a global phase cannot be measured, allows us to add a term eiφ that gives a relative phase between |0i and |1i. Thus, the state space of the single qubit can be represented using two variables, θ and φ:

|qi = cos θ 2

|0i + eiφ_sin θ

2

|1i . (3)

Now we can nicely represent the state space of a qubit by a sphere.

Figure 1: The Bloch sphere.

Things to note here are that measuring the x-, y- and z-component of the Bloch vector simultaneously is impossible. Also, regardless of the direction of the vector, measuring in the z-basis will only give the result 1 or -1, with probabilities cos2 θ₂ and sin2 θ₂ respectively.

We use quantum gates to perform operations on qubits, changing their state. They can either operate on multiple qubits, which will be discussed after this, or on a single qubit. If familiar with linear algebra, one should know about the Pauli matrices. The X matrix for example,

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acts as a X gate in a quantum circuit and when applied to a |0i state qubit, results in X |0i = " 0 1 1 0 # " 1 0 # = " 0 1 # = |1i . (4)

It represents the operation of either turning |0i into |1i or vice versa. Also, both the Y and Z matrices act as the Y and Z quantum gates:

Y = " 0 −i i 0 # , Z = " 1 0 0 −1 # . (5)

Another important quantum gate is the Hadamard gate, which allows a single qubit to go into a superposition of |0i and |1i, like in (2),

H = √1 2 " 1 1 1 −1 # . (6)

It performs the following transformations on the basis states of Z:

H |0i = √1

2(|0i + |1i) = |+i , H |1i = 1 √

2(|0i − |1i) = |−i , (7) where |+i and |−i are the basis states of X. Here, it performs a basis transformation from the basis of Z to that of X.

2.2 Multiple qubits and entanglement

The state of multiple qubits is the sum of tensor products of each qubits coefficients, for two qubits:

|ψi =X

i,j

cij|ii ⊗ |ji . (8)

The state space of the state obeys

|ψi ∈ (H2)⊗n (9)

where |ψi is the statevector of n qubits. This means that in order to do an operation on n qubits,

O |ψi = |ψ0i , (10) the dimension of O must be equal to the dimension of 2n, where n is the amount of qubits. For example, an operation O on 2 qubits, would have to be represented by a 4 × 4 matrix. The measurement rules,

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still uphold, and so does the normalisation condition X

i,j

|cij|2= 1. (12)

This is true for more than two qubits too. An example of an important 2 qubit gate, the CN OT gate, CN OT =        1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0        . (13)

It gives the following results when acting on qubits that are not in a superposition: CN OT |00i = |00i ,

CN OT |01i = |01i , CN OT |10i = |11i , CN OT |11i = |10i .

This can be seen as a bitflip under certain conditions. If the control qubit, is 0, it does nothing to the target qubit. However, if the control qubit is 1, it performs a bit flip on the target qubit. The control qubit here is the first in the ket, the target qubit being the second.

2.2.1 Entanglement

We saw the effects of a CN OT gate on 2 qubits that were not in a superposition, but what if they were? Say we applied a CN OT gate to 2 qubits in a |+0i state,

CN OT |+0i = CN OT [√1

2(|00i + |10i)] = 1 √

2(|00i + |11i). (14) We have now created a Bell state, which is interesting because it is an entangled state. This state can no longer be split in the product of two separate qubit states, which implicates that measuring one qubit, collapsing the superposition, will immediately tell us what the other qubit is, no matter how far apart they are [1].

3 Quantum computing in the NISQ era

A quantum computer essentially transforms a state |ψi into a state |ψ0i and does so by a series of operations on either one or multiple qubits in the (intermediate) state. These operations are programmed by way of a quantum circuit.

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3.1 Quantum circuits

Since a quantum computer essentially performs series of these operations on a starting state, a series of qubits in the |0i state, Fig. 2 is representative for all quantum circuits.

Figure 2: Quantum circuit representation of an operation on a state.

The U gate in Fig. 2 represents the accumulated unitary operation, consisting of all unitary operations done within the quantum circuit.

3.2 Native gates

In order to make sure that U in Fig. 2 can represent each and every possible operation on the initial state |0i⊗n, every quantum computing service supplies a set of universal gates, whose property is that any unitary operation U can be decomposed into a series of these gates. An example of a simple universal gate set is: [CN OT, H, T ]. Each Quantum Processing Unit (QPU) is equipped with a unitary gate set that is native to that QPU. The gates in that set are called native gates. Any unitary operation U , is decomposed into these native gates.

3.3 The NISQ era

Quantum computing today finds itself in the Noisy Intermediate-Scale Quantum (NISQ) era [8]. The QPUs that exist today are not fault-tolerant devices. This would require a lot of qubits, to be used for quantum error correction, since most quantum algorithms are very sensitive to noise. Today’s QPUs only sustain a couple dozen qubits (Intermediate-Scale) and have to directly use imperfect physical qubits. They will not be able to perform extensive error correction and remain noisy. Because of this, some things have to be taken into consideration when constructing quantum circuits. The three main problems when it comes to executing quantum circuits in the NISQ era are:

1. Decoherence 2. Gate noise

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4. Limited number of qubits

Decoherence is the phenomenon that causes the quantum state to lose its ‘quantumness’ [9, 10]. It turns the quantum mechanical probabilities of a system into classical probabilities. This happens because of the many random interactions the system has with its environ-ment, like changing magnetic or electrical fields, radiation or cross talk between qubits. After decoherence a quantum system will no longer be able to interfere with itself, losing its ‘quan-tumness’.

Then there is gate noise, of which there are two types; coherent and incoherent gate noise [11]. The first perturbes a unitary operation, but preserves the pure state of the wavefunction, so instead of |ψi 7→ U |ψi, |ψi 7→ U0|ψi, where U 6= U0. The chance of this happening is generally higher for multi qubit gates. There is also incoherent gate noise, which is a result of imperfect isolation (see previous paragraph on decoherence).

Measurement noise encompasses multiple phenomena, but the two most important are; classi-cal readout error and qubit decay during readout error [11]. The former is just noise resulting from the imperfections of the systems responsible for measuring and results in unintentional bit flips. The latter has the same effect on the outcome, that of a unintentional bit flip, but is caused by the duration of a measurement, which, if too long, allow states that behave like |0i states to collapse into states more closely resembling |1i states or vice versa.

These forms of error, the lack of qubits to correct for them, together with the lack of control we have over them, make it harder to construct and execute quantum circuits on any QPU and still get results that are useful. Generally, one should:

1. Limit the number of native gates.

2. Apply the circuit to the qubits that result in the highest possible circuit fidelity. 3. Limit the amount of multi-qubit native gates.

Operations on different qubits can be done simultaneously, but still every operation has a chance of coherent gate noise. Limiting the amount of native gates also means that two qubit gates should be done on neighbouring qubits, avoiding extra SWAP gates. The fact that operations can be done simultaneously on different qubits must be exploited as much as possible, reducing computation times and thus the effects of decoherence. Also, each qubit, or qubit pair has different fidelities for native gates, which should be taken into consideration when choosing qubits to apply a quantum circuit to. Multi qubit gates generally have a higher chance of perturbation, so minimise the use of them.

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4 The Variational Quantum Eigensolver

The Variational Quantum Eigensolver (VQE) is a classical-quantum hybrid algorithm, which is used to find an upper bound of the lowest eigenvalue of a Hamiltonian. This means that it is able to estimate the ground state energy of a system. Now why is the VQE so powerful? And what is its role in the possibilities within the NISQ era of quantum computing?

1. Finding the ground state is really useful, but very difficult, even for supercomputers. In both physics and chemistry, finding the ground state of a system, whether it is an atom, a molecule or protein, is fundamental. In chemistry this is used to determine useful qualities, such as reaction rates or binding strengths. In physics, it is used to estimate the ground state of quantum mechanical models for quantum magnetism for example. 2. The VQE was designed specifically to function despite of the shortcomings of the NISQ

era. The VQE is useful because it does not crack under accumulated noise [12].

4.1 Variational principle

So now that we know what a VQE is useful for, how does it actually work? The first part to understand, and coincidentally the concept the first letter in VQE stands for, is the variational principle. Say we have a system, with a certain Hamiltonian H. That operator H contains all the possible energies the system can have. The Hamiltonian and its eigenvalues have the relation

H |ψλi = λ |ψλi , (15)

where λ represents one of these eigenvalues, and |ψλi the system’s eigenstate corresponding

to that eigenvalue. The problem here is that it is really hard to know what the eigenstate or its corresponding eigenvalue is. So how do we at least find a good estimation? Consider how to even find the energy of a system,

hψ| H |ψi = E(ψ), (16) where E(ψ) is the energy corresponding to the state |ψi. Since the ground state energy is found with the state corresponding to the lowest energy the system can have, we know:

E(ψ) ≥ E0, (17)

where E0 is the ground state energy. This means that any time we do a measurement of

the energy according to (16) we set an upper bound for the systems ground state energy. Now this does not say much about the ground state energy of the system if only done once. Especially since the number of eigenvalues scales exponentially with the size of the system.

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To make the search for a good ground state energy estimation manageable, the VQE comes in.

4.2 Quantum-classical hybrid

The VQE is essentially an optimisation algorithm with state preparation and energy mea-surements on a QPU. It repeats the cycle in Fig. 3, until it finds a local minimum.

Figure 3: Visual representation of the optimisation cycle of a VQE.

This is, of course, a very blunt explanation of the VQE, but it helps to understand its process. Basically, in order to find an estimation of the ground state energy of a system using the VQE, first a state is prepared. The set of parameters and the way they are applied in the ansatz determine the states that can be prepared. The expectation value of the Hamiltonian is then measured using repeated state preparation. An optimisation algorithm is then used to steer the parameters until they produce a state corresponding to a local minimum, which gives us an upper bound for the ground state energy.

4.3 Quantum mechanical part

As stated before, the VQE is a quantum-classical hybrid algorithm. First, let us take a look at the quantum mechanical part of the algorithm, which consists of two parts

1. The preparation of the state. 2. Measurement of the energy.

We start of with an initial state, which can be anything. The choice of which state is certainly important, but this will be discussed later on. As mentioned before, the problem with finding a good ground state energy estimation with the help of the variational principle is that there are too many possible energies to find it by trial and error. The ground state energy lives in the Hilbert space of the Hamiltonian and to find it in a more effective manner, an ansatz is applied to an initial state. The ansatz is a way to make the state preparation dependent on the parameters and structure that define the ansatz.

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Then, the expectation value of the Hamiltonian, corresponding to the prepared state, is measured. On a real QPU it is not possible to directly measure the energy of a state, but it is possible to estimate it. This gives us our first upper bound of the ground state energy, according to (16). But is this way of searching for the ground state not still trial and error?

4.4 Classical part

This is where the classical part of the algorithm comes in. Since, as discussed before, the ansatz’s function is to parameterise quantum state preparation, we now not only have our first upper bound for the ground state energy, we also have the set of parameters that gave us this upper bound. The problem we are now left with is basically a classical optimisation problem. An optimisation algorithm is used to find the set of parameters that prepare the state corresponding to the lowest estimation of the Hamiltonian. Gradient-free algorithms are customary, since gradient based methods are sensitive to noise. Recently, however, methods have been proposed to create a Quantum Natural Gradient, which makes it possible to use first order gradient optimisation algorithms [13].

4.5 Choosing an ansatz

We want the ansatz to be as shallow as possible, while still spanning enough of the Hilbert space that the desired solution lives inside it. The trade-off here is that too shallow of an ansatz might not be able to ever produce a good estimation of the ansatz. While an ansatz with a lot of parameterised gates will suffer more from the limitations of the NISQ era, making the results increasingly untrustworthy. Also, an ansatz with more parameters makes the optimisation process more demanding.

There are many ways to construct an ansatz. Two important ones are the Hardware Efficient Ansatz (HEA) [14] and the Hamiltonian Variational Ansatz (HVA)[15, 16]. The HEA is designed to work on specific hardware, meaning it is constructed with gates native to a QPU. While it can be very expressive, it is not very efficient in finding the ground state energy since it still samples a lot of the Hilbert space. Also, HEAs suffer from barren plateaus [17], which are flat spots in Hilbert space that make it very hard for an optimisation algorithm to know what way to change the parameters in order to move towards the desired minimum.

An alternative is the HVA. This ansatz is more problem specific, and was shown to be more effective than the hardware efficient ansatz [15]. It allows for more efficient optimisation of pa-rameters, while still being very expressive. It is based on a scheme a many body Hamiltonian

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is expressed as a linear combination of smaller Hamiltonians,

H =X

s

Hs, (18)

where each Hs is one of these smaller Hamiltonians. Each Hs has its own layer of ansatz

gates, so that the circuit depth of the ansatz is s. The downside is that a HVA is not based on a QPUs native gates. This generally means the circuit depth is higher than that of a HEA, making it more prone to errors from accumulated gate noise and decoherence.

5 The XX-model

The XX-model is a quantum model, a limiting case of the Heisenberg spin model, where the spins only have x- and y-coupling. The Hamiltonian for the XX-model is:

HXX = X ⊗ X + Y ⊗ Y, (19)

where the X and Y are Pauli matrices. Confusingly, the XX-model is also sometimes called the XY -model.

6 Quantum circuit for a VQE algorithm to study the

XX-model

As discussed in the section on quantum computing in the NISQ era, there are some things to take into account when designing quantum circuits for a QPU. Even though the VQE is quite resilient to especially gate noise, one still has to follow these guidelines. Every QPU has a different set of native gates. This, together with the fact that each ansatz, suited for some problem, requires a different set of logic gates, means that for each ansatz, there is a QPU with native gates best suited for that ansatz.

6.1 Choosing a QPU

In this section two quantum computing hardware providers, IBMQ and Rigetti, and their QPUs, will be examined with these considerations and the problem at hand in mind and their differences and (dis)advantages will be discussed in order to conclude which suits our problem best.

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6.1.1 IBM Quantum Experience

IBMQ has U 1(λ), U 2(φ, λ) and U 3(θ, φ, λ) as its single qubit native gates and CN OT as its two qubit entangling native gate:

U 1(λ) = 1 0 0 eiλ ! , U 2(φ, λ) = √1 2 1 −eiλ eiφ ei(λ+φ) ! , (20) U 3(θ, φ, λ) = cos θ 2

−eiλ_sin θ 2

eiφsin θ₂ cos θ₂ ei(λ+φ)

! , CN OT =        1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0        . (21)

Any circuit written for IBMQ will be be decomposed into a series of these unitary gate operations.

6.1.2 Rigetti

Rigetti’s set of native gates consists of:

RX =

cos(φ₂) −i sin(φ₂) −i sin(φ₂) cos(φ₂)

! , RZ = cos(φ₂) − i sin(φ₂) 0 0 cos(φ₂) + i sin(φ₂) ! , (22) CZ =        1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 −1        , XY =        1 0 0 0 0 cos(φ₂) i sin(φ₂) 0 0 i sin(φ₂) cos(φ₂) 0 0 0 0 1        . (23)

The XY gate, or parameterised iSWAP gate, is especially interesting, since it is equal to the HVAs ansatz gate for the XX-model, up to some phase, which we can ignore. Its expression is:

UH = eiθH, (24)

where UH is the ansatz gate and H is a term of the Hamiltonian (18) of the system. For the

XX-model it is: UH = eiθ(XX+Y Y ) =        1 0 0 0 0 cos(2θ) i sin(2θ) 0 0 i sin(2θ) cos(2θ) 0 0 0 0 1        . (25)

This means that in order to construct an ansatz gate with Rigetti’s native gates, one param-eterised iSWAP gate is sufficient. IBMQ has no gates resembling (25) and will decompose into:

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Figure 4: Visual representation of the decomposition of UH for the hardware of IBMQ.

From this we can conclude that Rigetti here is preferable, since each ansatz gate only requires one 2-qubit parameterised iSWAP gate, while for IBMQ, each ansatz gate will be decomposed into six single qubit U3 gates and two 2-qubit CN OT gates.

6.2 The ansatz

In the previous section it is shown that an HVA with Rigetti as QPU would require just one native gate per ansatz gate. The HVA splits the Hamiltonian into parts (18) and for the XX-model, this means that H = H1+ H2 [7], where:

H1= L/2−1 X i=1 S2i· S2i+1, H2= L/2 X i=1 S2i−1· S2i, (26)

for an open chain. For a periodic chain, H1 is:

H1= L/2

X

i=1

S2i· S2i−1. (27)

Since the Hamiltonian is split in two parts, two layers of ansatz gates are required, one for H1

and one for H2, to add up to one ansatz cycle, see Fig. 6. The ansatz cycle can be applied

multiple times (increasing expressiveness), each time increasing the number of parameters by p . The number of parameters in the ansatz is:

Np = ncycle· p, (28)

where ncycle is the amount of times the ansatz cycle is repeated, each time increasing the

circuit depth by two, since each ansatz gate requires just one native parameterised iSWAP gate. In this case, the ansatsz both gains the advantages of an HVA, as well as the advantages of the HEA.

6.3 Initial state

Qubits usually start out in the |0i state, which is not an ideal state to apply the ansatz to. For a HVA we choose the initial state to be the ground state of H1 (26, 27) [16], which is a

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|01i − |10i √

2

⊗L₂

, (29)

where L is the length of the spin-1/2 chain.

Figure 5: Composition of native gates that returns a Bell singlet. The X gate is also native, since it is Rx(π). Also, for both the hardware suppliers |qni = |0i

Seen in Fig. 5 is the Bell quantum circuit for the Bell singlet preparation, which increases circuit depth by three, and requires two single qubit gates and one parameterised iSWAP gate. Optionally, both the XY and Rz gate can be parameterised.

6.4 Running Rigetti’s Grove VQE algorithm

In this thesis a VQE algorithm, provided by Rigetti’s Grove (an open source library that contains quantum algorithms) is used [11]. It requires a specific way of implementing the elements of the algorithm. To reiterate, the Hamiltonian of the system, a parameterised state and an optimisation algorithm. They are implemented in the following way:

1. The Hamiltonian as a ‘Pauli’ sum, a sum of Pauli terms (Pauli words). 2. A parameterised circuit, representing the parameterised state.

3. An optimisation algorithm.

Firstly, the Hamiltonian, represented by a Pauli sum. A Pauli sum is a sum of Pauli terms. Each Pauli term is a tensor product of Pauli matrices. The Pauli sum of the XX-model Hamiltonian is

L−1

X

n=0

XnXn+1+ YnYn+1 (30)

for an open chain. L here is the length of the chain. The periodic chain Pauli sum has an extra XLX0+ YLY0 term.

Then, the parameterised circuit. It contains the initial state and the ansatz, as shown in Fig. 6.

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(a) Open (b) Periodic

Figure 6: An example of a quantum circuit for the VQE, where L = 4 and ncycle = 1. Both for an open chain (a) and a periodic chain (b). Before the first dashed line, the initial Bell state preparation. Between the dotted lines, one ansatz cycle, which can be repeated.

Finally, the optimisation algorithm. The optimal algorithm is problem dependent and requires both trial and error and tuning [18]. Here we went with Powell’s optimisation algorithm, mainly because it needs less iterations for each run, and the amount of runs we can do is limited. Also, we found its optimisation time was lower for our specific problem.

6.5 Our quantum circuit

Our circuit is designed to run on Rigetti’s Aspen-9 QPU [19], its topology is shown in Fig. 7.

Figure 7: The topology of the Aspen-9 chip, where each number represents one qubit. The red line represents the longest possible chain of qubit pairs.

Now, every quantum circuit should be executed on the qubits that have the highest average fidelity for the gates that circuit is made up of, in our case mostly parameterised iSWAP gates. This is something a “PARTIAL” initial rewiring of the quantum circuit claims to achieve. One can also initially rewire the circuit “NAIVE”, which means that there will generally be no mapping between the logical qubits and the physical ones [20], provided that operations within the quantum circuit are possible within the topology of the QPU.

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7 Results

7.1 QVM

In order to debug the quantum circuit, the VQE is tested on a Quantum Virtual Machine (QVM) first. In this case a 9 qubit square grid, on which an 8-qubit chain was constructed [21]. The QVM simulates the unitary evolution of a wavefunction with classical control. In Fig. 8 we see the results of the VQE on the QVM without a noise model, where for each ncycle the VQE is run 10 times and the lowest value taken as the best ground state energy

estimation.

(a) Open (b) Periodic

Figure 8: Logarithmic plot of the relative error of the VQE’s estimation of the ground state energy compared to its exact value for the open and periodic chain as a function of ncycle. E0,ex is the exact ground state energy and E0,V QE is the VQE’s estimation of the ground state energy

We see in Fig. 8 that the relative error to the exact ground state energy of the ground state energy estimation of the VQE decreases exponentially as a function of ncycle. The exponential

evolution of the relative error as a function of ncycle is what is interesting here.

It is possible to apply a noise model to the QVM to debug noise related errors. This noise model incorporates decoherence, consisting of amplitude damping and dephasing, gate noise and asymmetric readout errors [22]. However, this simulation turned out to be too demanding for my laptop to do within a couple of days.

7.2 QPU

The quantum circuit, as shown in Fig. 6 turned out to not be possible, because of compati-bility issues between the Grove library and quilc, the compiler. These issues also applied for Aspen-9’s noise model on the QVM. However, we were able to conduct Bell singlet

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prepara-tion. The circuit for bell singlet preparation, see Fig. 5, was performed on all qubit pairs in the longest chain, see Fig. 7, from which the ‘success rate’ of Bell singlet preparation for one qubit pair was calculated. ‘Success rate’ here is the amount of qubit pairs whose measurement returned either |01i or |10i. For example, the measurement outcome

[10 11 10 01 0010 10 10 01 00 10 10 1011 00 ] (31) indicates a 10₁₅ × 100% = 67% ‘success rate’. This kind of measurement was performed 215

times for each initial rewiring, resulting in the following ‘success rates’. Initial rewiring ‘Succes rate’ (%)

“NAIVE” 66.3672 “PARTIAL” 66.2581

These percentages are of course also partially dependent on the readout fidelity of the qubits in the longest chain of qubit pairs on the Aspen-9, see fig. 7, which is 95.33% on average, and their Active Qubit Reset fidelity, averaging 96.88%.

8 Conclusions and future prospects

We highlighted the most prominent problems of the NISQ era, lack of qubits, decoherence, gate- and measurement noise. We therefore established some general guidelines for writing quantum circuits, both generally and specifically for a VQE. One should generally limit the number of gates, especially multi-qubit gates and stick to a QPU’s native gates as much as possible. When it comes to a VQE, a good ansatz should be both expressive and problem specific, like a HVA. The ansatz gates are also preferably native to a QPU, since otherwise increasing ncycle increases gate depth by more, and with it the effects of gate noise and

decoherence. Also, we showed that the accuracy of the VQE’s estimation of the ground state energy increases exponentially with ncycle.

Unfortunately, the envisioned algorithm was unable to run on the actual QPU. However, the success rate of Bell singlet preparation on the Aspen-9 QPU tells us that its performance is not yet ideal. The errors in the preparation could partially be absorbed by parameterising the angles in the parameterised iSWAP and Rz gates.

To achieve better results, either the software, hardware or both, have to improve. Regarding software, quantum algorithms have to become more resilient to the shortcomings of the NISQ era, similarly to the VQE. Also, it would not hurt to have more QPUs available to choose

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from. More QPUs means a wider selection of native gate sets and topologies. The former allows for problems to be solved more efficiently and more natively to a QPU, improving results. As for topologies, the more of them are available, the more likely the topology of some model will map to the topology of a QPU easily, without having to incorporate too many SWAP gates.

On the other hand, the limitations of the NISQ era are ones to be resolved via improvement of hardware too. Hardware improvements have far from reached their limits. Systems could still be isolated better from their environments, increasing coherence times. Decreasing the execution time of quantum gates and increasing their fidelities would be an improvement too.

These improvements will make it so that more complex models, think higher spacial dimension lattices or grids, with more ansatz gates, will still yield useful results on QPUs. An example of where this would be useful is the Kagom´e lattice according to the Heisenberg model. This can be helpful in the debate between long range magnetic order and spin liquids, for different topological configurations.

9 Acknowledgements

This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. These resources include access to Rigetti’s Quantum Computing Service.

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Variational Quantum Eigensolver for the XX-model