Combining the Transcorrelated Method with Adaptive Ansatzes for Near-term Quantum Computation Transcorrelated Adaptive Variational Quantum Imaginary Time Evolution Master’s thesis in Theoretical Chemistry Erika Magnusson DEPARTMENT OF CHEMISTRY AND CHEMICAL ENGINEERING CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2023 www.chalmers.se www.chalmers.se Master’s thesis 2023 Combining the Transcorrelated Method with Adaptive Ansatzes for Near-term Quantum Computation Transcorrelated Adaptive Variational Quantum Imaginary Time Evolution Erika Magnusson Department of Chemistry and Chemical Engineering Division of Physical Chemistry Rahmlab group Chalmers University of Technology Gothenburg, Sweden 2023 Combining the Transcorrelated Method with Adaptive Ansatzes for Near-term Quan- tum Computation Transcorrelated Adaptive Variational Quantum Imaginary Time Evolution Erika Magnusson © Erika Magnusson 2023. Supervisor: Werner Dobrautz, Department of Chemistry and Chemical Engineering Examiner: Martin Rahm, Department of Chemistry and Chemical Engineering Master’s Thesis 2023 Department of Chemistry and Chemical Engineering Division of Physical Chemistry Rahmlab group Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Cover: A selection of representative results for TC-AVQITE along with modelled systems. For more information, see chapter 4. Typeset in LATEX Printed by Chalmers Reproservice Gothenburg, Sweden 2023 iv Combining the Transcorrelated Method with Adaptive Ansatzes for Near-term Quan- tum Computation Transcorrelated Adaptive Variational Quantum Imaginary Time Evolution Erika Magnusson Department of Chemistry and Chemical Engineering Chalmers University of Technology Abstract The fundamental problem of quantum chemistry is that the cost of solving the electronic Schrödinger equation scales exponentially with system size. One way to potentially circumvent this scaling is using quantum computers, whose properties might enable an exponential computational speedup. However, near-term quantum computers struggle to compete with conventional quantum chemistry methodology. Near-term quantum processors are both small regarding the number of available qubits, and the number of operations that can be run on them is limited by sys- tematic noise. Despite this, various methods attempt to find quantum advantage in the current noisy intermediate-scale quantum (NISQ) regime. Variational quan- tum imaginary time evolution iteratively moves toward the energy ground state by evolving the quantum state in imaginary time. By splitting the evolution into steps and iterating, the necessary circuit depth is limited. This thesis evaluates the combination of variational quantum imaginary time evo- lution with two approaches aimed at decreasing quantum costs: the transcorrelated (TC) method of Boys and Handy and the adaptive ansatzes of Grimsley et al. The resulting method, transcorrelated adaptive variational quantum imaginary time evolution (TC-AVQITE), is evaluated through simulations of near-term quantum devices. The results for small systems (H2, quadratic H4, and lithium hydride) indicate that the combination works well, especially for increasing system sizes. This development takes us a step closer to chemically relevant calculations on quantum computers. Keywords: Quantum computer, Quantum chemistry, NISQ, QITE, VarQITE, ADAPT, ADAPT-VarQITE, Transcorrelated method. v Acknowledgements I want to thank Gomes et al. for allowing me to clone the code used in your paper! How lucky I am to have a supervisor with contacts; writing AVQITE from scratch would have taken quite a lot of time. Speaking of supervisor, I want to thank Werner Dobrautz for being a great help during all steps of the process! As a natural continuation of this, I want to thank Martin Rahm and the rest of the group for company and feedback. Finally, I also want to thank my opponent Oskar Olander. Erika Magnusson, Gothenburg, June 2023 vii Contents 1 Introduction 1 2 Theory 3 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 The Electronic Structure Problem . . . . . . . . . . . . . . . . 3 2.1.2 Why Quantum Computing? . . . . . . . . . . . . . . . . . . . 4 2.1.3 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Basis sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.3 Near-term approaches . . . . . . . . . . . . . . . . . . . . . . 7 2.2.4 Variational quantum eigensolver . . . . . . . . . . . . . . . . . 7 2.3 Quantum Imaginary Time Evolution . . . . . . . . . . . . . . . . . . 8 2.3.1 Variational QITE . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Adaptive Ansatzes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.1 Fermionic-ADAPT-VQE . . . . . . . . . . . . . . . . . . . . . 10 2.4.2 Qubit-ADAPT-VQE . . . . . . . . . . . . . . . . . . . . . . . 11 2.4.3 ADAPT-QITE – AVQITE . . . . . . . . . . . . . . . . . . . . 11 2.5 Transcorrelated Method . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Methods and Implementation 15 3.1 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.1 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.2 Basis set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.3 Operator pools . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.4 Hamiltonian generation . . . . . . . . . . . . . . . . . . . . . . 16 3.1.5 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Modelled systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Results and Discussion 21 4.1 Transcorrelated advantage . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Robustness of the algorithm . . . . . . . . . . . . . . . . . . . . . . . 27 4.2.1 Operator addition parameters . . . . . . . . . . . . . . . . . . 27 4.2.2 Operator addition criterion . . . . . . . . . . . . . . . . . . . . 28 4.3 AVQITE vs VarQITE . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ix Contents 5 Conclusion 33 Bibliography 35 A Additional results I A.1 Operator pool evaluation . . . . . . . . . . . . . . . . . . . . . . . . . I A.2 Mapping evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . II A.3 Condition numbers of A-matrix during AVQITE and VarQITE com- parison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II x 1 Introduction In quantum chemistry, the aim is to calculate atomic and molecular properties as faithfully as possible. From these fundamental calculations, one can obtain the ground state energy for the quantum system, from which chemically relevant proper- ties can be derived. The usual approach involves simulating the elementary particles in the molecules by making relevant approximations, and then using methods such as density functional theory or coupled cluster approaches. Here, a new method is gradually appearing – simulating chemical systems on a quantum computer. This method’s promise is great, as the fundamental properties of quantum computers al- low us to bypass that the wave function is exponentially scaling with particle number and system size. However, current quantum computers have many issues, such as systematic noise from imperfect quantum gates or a limited number of quantum bits (qubits). Despite the limits of current quantum computers, several methods are developed for quantum computation. Furthermore, there are also methods developed that cut down on quantum costs. Two of these are the transcorrelated method of Boys and Handy [1], as well as adaptive ansatzes developed by Grimsley et al. [2]. In this thesis, a combination of these two methods for cutting quantum computation costs is evaluated. Specifically, we want to investigate whether this combination outperforms simulations with only one of the methods and if the methods work well together. Quantum chemistry is a fundamental field whose developments could have long- reaching impacts. As quantum computation could theoretically let us calculate fundamental properties of molecules, it could in the future come to be used for developing new medicines or other useful chemical compounds. This could then have wide-reaching consequences for society at large. As it is now, a quantum revolution is not yet in sight, but by continued research it may one day be reached. 1 1. Introduction 2 2 Theory The theory chapter will start with the fundamental problem – the electronic struc- ture problem. After that, quantum computing is introduced and motivated, and the specific methods used in the thesis are detailed. 2.1 Background 2.1.1 The Electronic Structure Problem Fundamentally, computational quantum chemistry is all about the problem of cal- culating the states of electrons in atoms and molecules. This problem is known as the electronic structure problem. Determining the features of the electronic energy surface, i.e. the collective behaviour of the electrons, is important for several prop- erties such as reactivity and reaction rates, which motivates why the problem should be studied [3]. The problem can (in most cases) essentially be reduced to solving the stationary non-relativistic Schrödinger equation for various molecules, i.e., Ĥm(r⃗)Ψ(r⃗, t) = EΨ(r⃗, t), (2.1) where the molecular Hamiltonian in atomic units is given as Ĥm = − ∑ i ∇2 i 2︸ ︷︷ ︸ KE of e − ∑ I ∇2 I 2MI︸ ︷︷ ︸ KE of n − ∑ i,I ZI |r⃗i − R⃗I |︸ ︷︷ ︸ e-n attraction + 1 2 ∑ i ̸=j 1 |r⃗i − r⃗j|︸ ︷︷ ︸ e-e repulsion + 1 2 ∑ I ̸=J ZIZJ |R⃗i − R⃗j|︸ ︷︷ ︸ n-n repulsion . (2.2) Here, the uppercase indices go over the nucleons (concisely written as n in the equation), while the lowercase indices go over the electrons (e). The significance of each term is described under them – kinetic energies (KE), attraction, and repulsion. The charge and mass numbers Z and M are given in units of electron charges and masses. Under the Born-Oppenheimer approximation, we can decouple the electronic and nuclear Hamiltonians by assuming the nuclei to be stationary. The electronic Hamil- tonian then becomes Ĥ = − ∑ i ∇2 i 2︸ ︷︷ ︸ KE of e − ∑ i,I ZI |r⃗i − R⃗I |︸ ︷︷ ︸ e-n attraction + 1 2 ∑ i ̸=j 1 |r⃗i − r⃗j|︸ ︷︷ ︸ e-e repulsion . (2.3) 3 2. Theory It is possible to completely reformulate this Hamiltonian into the so-called second quantisation (or, more descriptively, the occupation number representation) by pro- jecting onto a basis set that spans the relevant Fock space. This basis set can be chosen freely if it spans the relevant space. There are several options, each with advantages and drawbacks (more about basis sets in Section 2.2.2). After choosing a basis set, one can construct creation and annihilation operators a† p, ap, which add and remove electrons in the spin orbital p respectively. These are constructed in accordance with the anti-commutative property of fermions, {ap, aq} = {a† p, a † q} = 0, {ap, a † q} = δp q . (2.4) In the second quantisation, the electronic Hamiltonian thus looks like Ĥ = ∑ pq hq pa † paq︸ ︷︷ ︸ one-body terms + 1 2 ∑ pqrs V rs pq a † pa † qaras︸ ︷︷ ︸ two-body terms , (2.5) with hq p = ∫ dx⃗ϕ∗ p(x⃗) ( −∇2 2 − ∑ I ZI |r⃗ − R⃗I | ) ϕq(x⃗) (2.6) and V rs pq = ∫ dx⃗1dx⃗2 ϕ∗ p(x⃗1)ϕ∗ q(x⃗2)ϕr(x⃗1)ϕs(x⃗2) |r⃗1 − r⃗2| , (2.7) where the basis functions are ϕ(x⃗). To clarify, the indices p, q... here go through all spin orbitals. The second quantisation will be assumed throughout the rest of the thesis. The approach is a good match with quantum computing, as it often gives simpler Hamil- tonians and thus requires fewer qubits in general [4]. 2.1.2 Why Quantum Computing? Various methods exist for solving this problem, ranging from mean-field theories like Hartree-Fock to expensive ab initio methods. One new and promising method is using quantum computers, as their properties are particularly well suited for quantum simulation [3], which in the end, is what we want to do in computational chemistry. That they match so well is why the electronic structure problem is often thought to be one of the first useful applications of quantum computers [5]. As a field, quantum chemistry can benefit immensely from powerful quantum com- puters, which would let us solve problems which are intractable on normal comput- ers. In particular, problems that explicitly require the wave function due to the simulated properties, problems with a high degree of entanglement, and problems involving highly correlated systems, could benefit immensely [5, 6]. Furthermore, other fields, like materials science, can also benefit from the methods developed for quantum chemistry [3]. For example, a highly correlated system in physics would be high-temperature superconductors, which are studied in the context of the Hubbard model. The potential of quantum simulation is clear to see – and one day, it may become more accurate to theory than measurements of actual experiments. This would be revolutionary and enable designing useful molecules from scratch [5]. 4 2. Theory 2.1.3 Caveats There are, of course, a few caveats to consider. Firstly, for the truly groundbreak- ing quantum simulation, it is estimated that orders of magnitude more qubits are needed than what are available today, plus error correction and lower error rates [5]. Thus, in the near term, quantum computers cannot quite deliver what is prophesied, but could still outperform classical options for highly correlated systems [6]. Simu- lations have shown that certain approaches can reach competitiveness with classical methods with only a few qubits [7]. Secondly, quantum computing with today’s quantum hardware is difficult, and most computations done today correspond to simulations done decades ago on classical hardware. The calculations done today are not quite yet pushing the limits of science as it were, but are more proof of concept. Thirdly and finally, not everything is suited for quantum computers. Quantum advantage – when a quantum computer outperforms a classical computer – is not universal, but problem specific. An arbitrary simulation will therefore not necessar- ily be sped up when done on a quantum computer, and one must instead identify which problems are worth considering [6, 3]. Generic ground state determination lies in complexity class QMA, which is a class of problems whose relative efficiency of exponential speedup on a quantum computer is not theoretically guaranteed [8, 9]. Fortunately, recent studies suggest that the problem as addressed in quantum chem- istry can be made BQP-complete, which thus in turn suggests quantum advantage [10, 11]. 2.2 Quantum Computing As mentioned, quantum computing is an emerging alternative to classical compu- tation strategies. A large part of their benefit lies in the inherent advantage that quantum computers can store the entire wave function, which is exponentially scal- ing for classical computers. This is possible because N qubits can store 2N bits of information, circumventing the exponential scaling issue. There are two ways of us- ing quantum hardware to simulate real systems – analogue and digital [5]. Analogue quantum simulation would be done by constructing hardware mimicking a system of interest, while digital computation maps a problem to a more generic quantum computer. This thesis is strictly about digital quantum computation. 2.2.1 Encoding To do a (digital – from now on omitted) quantum computation, one must some- how transcribe the problem to the quantum computer, which is not a trivial task. The largest complication is that all known quantum computers have distinguishable qubits, but electrons, which we wish to simulate, are indistinguishable in nature [6]. Several mappings exist, but the simplest one is the Jordan-Wigner (JW) mapping, where a qubit directly represents the occupation number of a spin-orbital. As an example, in the electronic structure problem, the Hartree-Fock-state for a two-level 5 2. Theory system with two electrons would be |ΨHF⟩ =̇ |0011⟩ , (2.8) where the registers from right to left are spin down ground state, spin up ground state, spin down excited state, and spin up excited state, respectively. This is one possible implementation of the qubit order for the JW mapping – one can as easily order them in some other way. This has no real impact on the simulated systems though, only on the algorithmic implementation. Exciting the spin-down electron in this implementation of JW mapping would give a† 1↓a0↓ |0011⟩ = |0110⟩ . (2.9) For an illustration of these two states, see Figure 2.1. Figure 2.1: An illustration of the two states represented in the JW mapping in equations (2.8) (left) and (2.9) (right). The advantage of the JW mapping is its simplicity. However, other, less straight- forward mappings are more efficient, such as parity mapping [12]. Amongst other features, the parity mapping makes use of inherent symmetries of the problem, in this case that the number of up- and down-spin electrons are conserved, which allows us to reduce the necessary number of qubits to encode the system by two compared to the JW mapping [12]. This is a particularly large advantage in near-term com- putation, as the number of qubits is limited. 2.2.2 Basis sets The mapping is not the only thing that needs to be considered; one must also consider which basis set is used in equation (2.5). As mentioned, these basis sets have various properties that make them attractive in certain scenarios but inadvisable in others. One relevant class of basis sets is the minimal ones, particularly the STO- nG family of basis sets. This stands for Slater Type Orbital, n Gaussians. Of these, STO-6G is both readily available and as accurate as possible while still being minimal. Due to this, it was used for all systems except the smallest, the hydrogen dimer. A minimal basis set is the smallest possible set of basis functions to describe a system. Another relevant class of basis sets are the Pople basis sets, which are split- valence basis sets that allow for greater flexibility in the Hartree-Fock optimisation. A Pople basis set was used for the hydrogen dimer – specifically, 6-31G. 6 2. Theory 2.2.3 Near-term approaches Once the electronic structure problem is mapped to the quantum computer, there are several algorithms for solving it. However, due to the size of near-term quantum computers, most methods are still inaccessible to us [6]. For the near-term devices, the focus is on methods that could work well on the noisy and small-scale hardware available – these devices are known as Noisy Intermediate Scale Quantum (NISQ) computers. These methods try to be NISQ-friendly by offloading parts of the com- putation to classical computers and only using quantum resources when necessary. As a result, circuit depth and coherence time requirements decrease. 2.2.4 Variational quantum eigensolver The most straightforward of these methods is the Variational Quantum Eigensolver (VQE). The variational quantum eigensolver works by preparing a trial state, an ansatz, dependent on a set of parameters θ⃗ on the quantum hardware. This ansatz is then used to calculate the expected energy, which according to the variational theorem will always be higher or equal to the ground state energy [13]. The pa- rameters are then passed to a classical computer, where an optimisation procedure follows. Afterwards, a new set of parameters are obtained, which will be used to prepare a new trial state. This procedure is repeated until a minimum is reached, which is hopefully the ground state energy. Figure 2.2 depicts the algorithm. Figure 2.2: An overview of the VQE algorithm. Taken with permission from Werner Dobrautz. While incredibly simple in idea and comparatively easy to execute, some issues must be considered. Firstly, the aforementioned ansatz needs a non-vanishing overlap with the actual ground state wave function, or the converged energy will not be the ground state energy [6]. One can intuitively see that the quality of the ansatz has a large impact, so one would want as physical of an ansatz as possible from this perspective. However, the state must also be comparatively easy to prepare on quantum hardware, or it will become too expensive for NISQ computers. Trying to 7 2. Theory balance this leads to the two classes of physically motivated and hardware-efficient ansatzes, where one property is prioritised over the other. A long-time friend of chemists is coupled cluster theory, which in the quantum computer is realised as the physically motivated unitary coupled cluster (UCC), often truncated at double excitations (UCCSD) [14]. An example of a hardware- efficient ansatz is the RY ansatz [15], which tries to be as minimal as possible to implement, but might not conserve physical properties such as the number of electrons during the optimisation. On their own, hardware-efficient ansatzes often lead to problems in the optimisation step of VQE, which makes them a comparatively poor match [16]. Finally, the VQE algorithm unsurprisingly fails if the Dirac-Frenkel variational prin- ciple (i.e., the “normal” variational principle) does not hold, for example if the Hamiltonian were to become non-Hermitian [17]. 2.3 Quantum Imaginary Time Evolution Quantum Imaginary Time Evolution (QITE) is a quantum computer implementa- tion of the concept of imaginary time evolution [18]. This is a nonphysical math- ematical trick used in various fields such as statistical mechanics, cosmology, and quantum mechanics [19]. Imaginary time evolution is a way of determining the ground state energy by replacing time in normal time evolution (i.e., dynamics sim- ulation) with imaginary time: t → iτ . Given an initial state at starting time |ψ(0)⟩, the normalised imaginary time evolution to imaginary time τ is |ψ(τ)⟩ = e−Ĥτ |ψ(0)⟩√ ⟨ψ(0)| e−2Ĥτ |ψ(0)⟩ . (2.10) As τ → ∞, the state is guaranteed to go to the ground state for the Hamiltonian Ĥ given that the initial state overlaps with the ground state [19]. An equivalent formulation to this is to solve the Wick-rotated Schrödinger equation instead [19]: ∂ ∂τ |ψ(τ)⟩ = −(Ĥ − Eτ ) |ψ(τ)⟩ , Eτ = ⟨ψ(τ)| Ĥ |ψ(τ)⟩ . (2.11) While the guarantee of convergence sounds tempting, there are implementation is- sues. It is sadly not straightforward to decompose this evolution into a sequence of unitary gates on a quantum computer, and is thus hard to implement as a circuit [19]. Furthermore, these circuits would likely be very deep and thus not NISQ-friendly. Compared to VQE, this is thus not something immediately useful. 2.3.1 Variational QITE However, there is a way to reformulate QITE in a variational manner [19]. The resulting algorithm, VarQITE, is a hybrid quantum-classical algorithm similar to VQE. In doing this, we sadly lose the strict guarantee of convergence due to approx- imation. Figure 2.3 depicts the algorithm. 8 2. Theory Figure 2.3: An overview of the VarQITE algorithm. Taken with permission from Werner Dobrautz. To get VarQITE, one approximates the starting state |ψ⟩ as an ansatz, depending on a set of parameters θ⃗, |ψ(τ)⟩ ≈ ∣∣∣ϕ(θ⃗(τ)) 〉 =̇ |ϕ(τ)⟩ . (2.12) The imaginary time evolution is then approximated using McLachlan’s variational principle [20], δ ∣∣∣∣∣ ( ∂ ∂τ + Ĥ − Eτ ) |ψ(τ)⟩ ∣∣∣∣∣ = 0, | |ψ⟩ | = √ ⟨ψ|ψ⟩. (2.13) By expanding in the parameter space and simplifying, one obtains the following differential equation: ∑ j Aij θ̇j = Ci, (2.14) where Aij = Re ( ∂ ⟨ϕ(τ)| ∂θi ∂ |ϕ(τ)⟩ ∂θj ) , Ci = Re ( −∂ ⟨ϕ(τ)| ∂θi Ĥ |ϕ(τ)⟩ ) . (2.15) A and C depend on the imaginary time τ . They can be interpreted as the metric A(τ), related to the quantum Fisher information matrix [21], and the gradient C(τ), respectively. It is possible to implement VarQITE in such a way that you do not need to measure the energy at every iteration, as long as you track these two objects (as in Figure 2.3) [7]. With these, one can approximate the imaginary time evolution iteratively with, for example, the Euler method [19]. Solving this requires inverting the metric matrix A, which is often ill-conditioned. This can be avoided with Tikhonov regularisation [19]. The reason why McLachlan’s variational principle is used rather than perhaps the more familiar Dirac-Frenkel variational principle is that it is the most appropriate 9 2. Theory for QITE [22]. This is, amongst other reasons, because it will always give real solutions given real parameters. Given complex parameters the principles are equiv- alent, but limiting ourselves to real wave functions is thus possible with McLachlan’s formulation [22]. An important measure is the McLachlan distance L2, defined as L2 = ∑ i,j Aij θ̇iθ̇j − 2 ∑ i Ciθ̇i + 2 Var(Ĥ). (2.16) The significance of the McLachlan distance is that it shows how accurate the simula- tion is – a smaller McLachlan distance means a smaller difference between variational and normal QITE, and thus the “optimal” path to the ground state. Having a high McLachlan distance can lead to “walking off the path” to the ground state, and thus ending up in a higher energy state. VarQITE has several advantages compared to VQE. Firstly, it avoids the optimisa- tion problems that plague VQE when the parameter landscape is mostly flat, and generally performs better than VQE in the presence of noise [19]. Furthermore, it is possible to apply to non-Hermitian problems, and implementation of it can be optimised for this [17]. Overall, it is thus a competitive alternative to VQE, and a NISQ-friendly way of handling non-Hermitian problems, if such should arise. 2.4 Adaptive Ansatzes Adaptive ansatzes are a class of ansatzes that are iteratively built for the problem to be simulated. The first adaptive ansatz was the Fermionic-ADAPT-VQE by Grimsley et al. [2], and several versions have followed from it, optimised for various purposes. The origin of adaptive ansatzes lies in the realisation that the choice of ansatz has a large impact on the quality of a variational quantum simulation – “a simulation is only as good as the ansatz” [2]. 2.4.1 Fermionic-ADAPT-VQE The original adaptive ansatz, ADAPT-VQE (short for Adaptive Derivative-Assembled Psuedo-Trotter), constructs an ansatz from a predefined operator pool consisting of UCCSD excitations, which are chosen depending on their contribution to the gradi- ent to get a maximally compact ansatz. The algorithm to develop an ansatz in ADAPT-VQE begins with a few preparatory steps: 1. Compute the one-body and two-body integrals in equations (2.6) and (2.7) on classical hardware. 2. Define the operator pool, for example, all UCCSD excitations. 3. Initialise the qubits at some reference state, for example, the HF state. This is then followed by repeating the following steps until a specified tolerance is reached: 1. Prepare a trial state on the quantum hardware with the current ansatz. 2. Measure the commutator of each operator in the operator pool to get the gradient. 10 2. Theory 3. If the norm of the gradient is smaller than the tolerance, exit. Otherwise, find the operator in the pool with the largest contribution to the gradient and add it to the ansatz. 4. Reoptimise all parameters and restart. The reason for measuring the gradient is to find the best operator to add to the ansatz, as the operator with the largest gradient will have the largest impact on energy minimisation. The ADAPT-VQE successfully decreases the circuit depth compared to including the entire operator pool, but the cost paid is more measure- ments. This is good for NISQ hardware, where circuit depth is one of the limiting factors. Interestingly enough, the ADAPT-VQE ansatz performs better than includ- ing all operators in the pool regarding the accuracy, while also minimising parameter count. [2] 2.4.2 Qubit-ADAPT-VQE An important realisation was made by Tang et al. [23]: A parameter-efficient pool is not necessarily gate-efficient. Thus, Fermionic-ADAPT-VQE does not minimise the ansatz depth. Furthermore, it is unclear how to construct the operator pool or whether the constructed ansatz will be good enough for convergence. These are the main factors driving the development of a hardware-efficient ADAPT algorithm called Qubit-ADAPT-VQE. The main difference lies in the choice of operators in the operator pool – Qubit- ADAPT-VQE has replaced the fermionic operators in its predecessor with a min- imised set of Pauli strings, series of repeating Pauli gates. Thus, the amount of gates is decreased. The pool is chosen to be as small as possible while still guaranteeing completeness, i.e., that it can adequately describe the ground state. This minimum pool size is shown to scale linearly with the number of qubits. The cost for minimising circuit depth is that the number of parameters is no longer strictly minimised, which naturally leads to larger parameter counts than fermionic ADAPT. [23] 2.4.3 ADAPT-QITE – AVQITE AVQITE is what you get if you apply the adaptive ansatz formalism to variational quantum imaginary time evolution, which was done by Gomes et al. [24]. However, as VQE and VarQITE are quite different, there are many differences in implemen- tation specifics. The initial steps are similar to a normal VarQITE calculation but with an arbitrary first ansatz. During the evolution, the number of operators is allowed to be ex- panded. This is done by appending operators after each imaginary time step if the conditions to grow the ansatz are met. In ADAPT-VQE, it is simple to decide when to append an operator and which operator to pick, but the same evaluation method does not work in AVQITE. Instead, the ansatz is grown by adding operators which minimise the McLachlan distance (Eq. (2.16)), thus keeping the evolution as close to normal QITE as possible. As an extension of this, operators are added when the McLachlan distance becomes too large. 11 2. Theory AVQITE can immediately be realised with a hardware-efficient operator pool like Qubit-ADAPT-VQE by, for example, choosing all the Pauli strings in the UCCSD ansatz as the operator pool. This pool was chosen by Gomes et al., whose results showed over one order of magnitude shallower circuits than the UCCSD ansatz. [24] 2.5 Transcorrelated Method The transcorrelated (TC) method was introduced by Boys and Handy [1], and is an explicitly correlated method based on factorising the electronic wave function in Jastrow form [25], |ψ⟩ = eJ |ϕ⟩ . (2.17) To be a proper Jastrow ansatz, J has to satisfy J = ∑ i