Variational quantum algorithms for nonlinear problems

Michael Lubasch1, Jaewoo Joo1, Pierre Moinier2, Martin Kiffner3,1, and Dieter Jaksch1,3

Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom1

BAE Systems, Computational Engineering, Buckingham House,FPC 267 PO Box 5, Filton, Bristol BS34 7QW, United Kingdom2 and

Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 1175433

We show that nonlinear problems including nonlinear partial differential equations can be effi-ciently solved by variational quantum computing. We achieve this by utilizing multiple copies ofvariational quantum states to treat nonlinearities efficiently and by introducing tensor networks asa programming paradigm. The key concepts of the algorithm are demonstrated for the nonlinearSchrodinger equation as a canonical example. We numerically show that the variational quantumansatz can be exponentially more efficient than matrix product states and present experimentalproof-of-principle results obtained on an IBM Q device.

Nonlinear problems are ubiquitous in all fields of sci-ence and engineering and often appear in the form ofnonlinear partial differential equations (PDEs). Stan-dard numerical approaches seek solutions to PDEs ondiscrete grids. However, many problems of interest re-quire extremely large grid sizes for achieving accurate re-sults, in particular in the presence of unstable or chaoticbehaviour that is typical for nonlinear problems [1–3]. Examples include large-scale simulations for reliableweather forecasts [4–6] and computational fluid dynam-ics [7–9].

Quantum computers promise to solve problems thatare intractable on conventional, i.e. standard classical,computers through their quantum-enhanced capabilities.In the context of PDEs, it has been realized that quantumcomputers can solve the Schrodinger equation faster thanconventional computers [10–12], and these ideas havebeen generalized recently to other linear PDEs [13–18].However, nonlinear problems are intrinsically difficult tosolve on a quantum computer due to the linear nature ofthe underlying framework of quantum mechanics.

Recently, the concept of variational quantum comput-ing (VQC) attracted considerable interest [19–32] forsolving optimization problems. VQC is a quantum-classical hybrid approach where the evaluation of the costfunction C(λ) is delegated to a quantum computer, whilethe optimization of variational parameters λ is performedon a conventional classical computer. The concept ofVQC has been applied, e.g., to simulating the dynamicsof strongly correlated electrons through non-equilibriumdynamical mean field theory [22, 23, 33, 34], and quan-tum chemistry calculations were successfully carried outon existing noisy superconducting [21, 25, 26] and ionquantum computers [28].

We extend and adapt the concept of VQC to solv-ing nonlinear problems efficiently on a quantum com-puter by virtue of two key concepts. First, we in-troduce a quantum nonlinear processing unit (QNPU)that efficiently calculates nonlinear functions of the form

F = f (1)∗∏rj=1(Ojf (j)) for VQC. Measuring the ancillaqubit connected to the QNPU as shown in Fig. 1(a) di-rectly yields the sum of all function values ∑kR{Fk},

0 3 6300

60000

0

2

4

FIG. 1. (a) Quantum network for summing a nonlinear func-

tion of the form F = f (1)∗∏rj=1(Ojf (j)). The ancilla qubit on

the top line undergoes Hadamard gates H and controls op-erations of the rest of the network via the control port (CP)and is measured in the computational basis. Starting fromproduct states ∣0⟩ of n qubits shown as thick lines, varia-

tional states ∣ψ(λ)j⟩ = Uj(λ)∣0⟩ representing the functions

f (j) are created and fed to the QNPU through input portsIP. The QNPU contains problem specific quantum networksdefining the linear operators Oj and has the output ports

OP. (b) Network U(λ) of depth d = 5 with n = 6. The valuesλ = {λ1,λ2, . . .} determine the form of the two-qubit gates.(c) Cost function C(λ) for the nonlinear Schrodinger equationEq. (1) for a harmonic potential V , a single variational param-eter λ, and g = 104 (solid line) and g = 10 (dashed line). Thearrows indicate optimal values λ = λmin. (d) Solutions ∣f(x)∣2for λ = λmin with N = 4 grid points and periodic boundaryconditions f(b) = f(a) for g = 104 (solid line) and g = 10(dashed line). The circles and squares are numerically exactresults. Experimental data in (c) and (d) was obtained – froma modified circuit – on IBM Q devices (see [35] for details).

where R{⋅} denotes the real part. The functions f (j) are

encoded in variational n-qubit states ∣ψ(λ)j⟩ = Uj(λ)∣0⟩

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created by networks of the form shown in Fig. 1(b). The

same function f (i) = f (j) may appear multiple times bychoosing Ui(λ) = Uj(λ). Second, we use tensor networksas a programming paradigm for QNPUs to create opti-mized circuits that efficiently calculate linear operatorsOj acting on functions f (j). In this way all quantumresources of nonlinear VQC scale polynomially with thenumber of qubits, which represents an exponential reduc-tion compared with some conventional algorithms.

The variational states ∣ψ(λ)j⟩ represent N = 2n values

of the functions f (j) which form a trial solution to theproblem of interest. The cost function C(λ) for nonlinearVQC is built up from outputs of different QNPUs thatare then processed classically to iteratively determine theoptimal set λ. Large grid sizes that are intractable on aconventional computer require only n ≳ 20 qubits which iswithin reach of noisy intermediate-scale quantum (NISQ)devices. In addition, the scheme is applicable to othertypes of nonlinear problems that can be solved via theminimization of a cost function C(λ) [36].

We demonstrate the concept and performance of non-linear VQC by emulating it classically for the canonicalexample of the time-independent one-dimensional non-linear Schrodinger equation

[−1

2

d2

dx2+ V (x) + g∣f(x)∣2] f(x) = Ef(x) , (1)

where V is an external potential and g denotes thestrength of the nonlinearity. We also implement nonlin-ear VQC for Eq. (1) on IBM quantum computers to es-tablish its feasibility on current NISQ devices. Proof-of-principle results are shown in Figs. 1(c) and 1(d) demon-strating excellent agreement with numerically exact solu-tions. The nonlinear Schrodinger equation and its gener-alizations to higher dimensions describe various physicalphenomena ranging from Bose-Einstein condensation tolight propagation in nonlinear media [37–42]. In particu-lar, below we consider Eq. (1) with quasi-periodic poten-tials V that are in the focus of current cold atom exper-iments [43] and that make Eq. (1) challenging to solvenumerically. The methods used for this equation hereare straightforwardly modified to handle other nonlinearterms and time-dependent problems as illustrated in [35]for the Burgers equation appearing in fluid dynamics.

The ground state of Eq. (1) can be found by minimizingthe cost function

C = ⟪K⟫c + ⟪P⟫c + ⟪I⟫c , (2)

where ⟪K⟫c, ⟪P⟫c and ⟪I⟫c are the mean kinetic, po-tential and interaction energies, respectively. In Eq. (2)⟪⋅⟫c denotes averages with respect to a single real-valued

function f (1) ≡ f on the interval [a, b] satisfying the nor-

malization condition ∫ ba ∣f(x)∣2dx = 1.In line with standard numerical approaches [44–46] we

apply the finite difference method (FDM) to Eq. (1) anddiscretize the interval [a, b] into N equidistant grid points

xk = a+hNk, where hN = `/N is the grid spacing, ` = b−ais the length of the interval and k ∈ {0, . . . ,N − 1}. Eachgrid point is associated with a variational parameter fkthat approximates the continuous solution f(xk) at xk.Furthermore, we impose periodic boundary conditions(i.e., fN = f0) and the normalization condition imposedon the continuous functions f translates to

1 = hN N−1∑k=0 ∣fk ∣2 = N−1∑

k=0 ∣ψk ∣2 , (3)

where ψk = √hNfk. Note that the condition on the set of

parameters {ψk} is independent of the grid spacing, andin the following we consider optimizing the cost functionwith respect to them.

All averages ⟪⋅⟫c in Eq. (2) can be approximated bycorresponding expressions of the discrete problem ⟪⋅⟫.We find [44–46] ⟪⋅⟫c = ⟪⋅⟫ + Egrid, where Egrid ∝ 1/N2

is the error associated with the trapezoidal rule whentransforming integrals into sums, and

⟪K⟫ = −1

2

1

h2N

N−1∑k=0 ψ

∗k (ψk+1 − 2ψk + ψk−1) , (4a)

⟪P⟫ = N−1∑k=0 [ψ∗kV (xk)ψk] , (4b)

⟪I⟫ = 1

2

g

hN

N−1∑k=0 ∣ψk ∣4 . (4c)

Note that ⟪K⟫ in Eq. (4a) uses a FDM representation ofthe second-order derivative in Eq. (1).

For evaluating the terms in Eq. (4) on a quantumcomputer we consider quantum registers with n qubitsand basis states ∣q⟩ = ∣q1 . . . qn⟩ = ∣q1⟩ ⊗ . . . ⊗ ∣qn⟩, whereqj ∈ {0,1} denotes the computational states of qubit j.Regarding the sequence q1 . . . qn = binary(k) as the bi-nary representation of the integer k = ∑nj=1 qj2n−j , weencode all N = 2n amplitudes ψk in the normalized state

∣ψ⟩ = N−1∑k=0 ψk ∣binary(k)⟩ . (5)

We prepare the quantum register in a variational state∣ψ(λ)⟩ via the quantum circuit U(λ) of depth d shown inFig. 1(b). We consider depths d∝ poly(n) such that thequantum ansatz requires exponentially fewer parametersN than standard classical schemes with N parameters.Note that the number of variational parameters scaleslike N ∝ nd for our quantum ansatz [35]. The powerof this ansatz is rooted in the fact that it encompassesall matrix product states (MPS) [47–50] with bond di-mension χ ∼ poly(n) [35]. Since polynomials and Fourierseries [51, 52] can be efficiently represented by MPS, thequantum ansatz simultaneously contains universal basisfunctions that are capable of approximating a large classof solutions to nonlinear problems efficiently. Further-more, we show below that the quantum ansatz is capableof storing solutions with exponentially fewer resources

3

FIG. 2. (a) QNPU circuit calculating the nonlinear term∣ψ∣4. The networks are shown for n = 3 and all input ports IP

are fed the same variational quantum states created by U(λ).IP2 is fed U(λ) and IP3 is fed U∗(λ). (b) QNPU circuit for

working out the potential energy term V ∣ψ∣2.

than the classically optimized MPS ansatz. Note that thenumber of variational parameters scales like N ∝ nχ2 forMPS of bond dimension χ [35].

Figure 2(a) demonstrates the basic working principleof the QNPU for the nonlinear term ⟪I⟫. The effect ofthe controlled NOT operations between pairs of qubitsis to provide a point-wise multiplication with the ancillathus measuring ∑k ∣ψk ∣4. In Fig. 2(b) we show the circuit

for measuring ⟪P⟫. The unitary V =O1 encodes functionvalues of the external potential V . A copy of ψ is effec-tively multiplied point-wise with the external potentialby controlled NOT gates to give ∑k Vk ∣ψk ∣2. Similarly,multiplying ψ with their shifted versions using adder cir-cuits (see [35] for details) allows evaluating the kineticenergy term.

The measured expectation value of the ancilla qubitis directly related to the desired quantities as ⟪I⟫ =g⟨σz⟩Ianc/2hN for the nonlinear term, ⟪P⟫ = α⟨σz⟩Panc forthe potential energy and ⟪K⟫ = (1 − ⟨σz⟩Kanc) /h2N for thekinetic energy [53, 54]. Furthermore, derivatives of thecost function, as required by some minimization algo-rithms [36, 55, 56], can be evaluated by combining theideas presented here with the quantum circuits discussedin [24, 30, 57, 58].

The unitary network V represents scaled function val-ues Vk of the external potential where ∑N−1

k=0 ∣Vk ∣2 = 1,

and α > 0 a scaling parameter such that Vk = αVk. Ef-ficient quantum circuits V for measuring ⟪P⟫ can besystematically obtained by establishing tensor networksas a programming paradigm. To this end we expandthe external potential in polynomials or Fourier series,V (x) ≈ ∑Jj cjbj(x), where bj(x) are basis functions andcj are expansion coefficients [35]. In the case of Fourierseries of order J , the approximate potential is representedby an MPS of bond dimension χ = J [51, 52]. Nextwe write the MPS in terms of n − ⌈logχ⌉ unitaries [59–61], where ⌈⋅⌉ is the ceiling function. Each of these uni-taries acts on 2χ qubits and can be decomposed in termsof elementary two-qubit gates [35, 62–65]. An upper

bound for the depth of the resulting quantum circuit isd> ≤ 9n[(23/48)(2χ)2 + 4/3] [64]. The depth thus scalespolynomially with the number of qubits n and with χ,and many problems of interest show an even more ad-vantageous scaling. For example, in the following weconsider the potential

V (x) = s1 sin(κ1x) + s2 sin(κ2x) (6)

and set κ2 = 2κ1/(1 + √5). This potential realizes an

incommensurate bichromatic lattice where the ratio s1/s2determines the amount of disorder in the lattice [66]. Thetrap potential V (x) in Eq. (6) is exactly represented byan MPS of bond dimension χ = 4. The depth of thecorresponding quantum circuit d = 5(n − 2) + 1 ≪ d> ismuch smaller than the upper bound [35].

Next we analyze the Monte Carlo sampling error [44]associated with the measurement of the ancilla qubit.We denote the absolute sampling error associated withquantity X by EXMC, and the corresponding relative erroris [35]

εPMC = EPMC⟪P⟫ = CP 1√M

, (7a)

εKMC = EKMC⟪K⟫ ≈ CK N

Nmin

1√M

, (7b)

εIMC = EIMC⟪I⟫ ≈ CI N

Nmin

1√M

. (7c)

In this equation, we assume N ≥ Nmin and Nmin = `/`min

is the minimal number of grid points for resolving thesmallest length scale `min of the problem. The parame-ters CX in Eq. (7) are of the order of unity [35], and allsampling errors decrease with the number of samples Mas 1/√M . While the relative error associated with thepotential term in Eq. (7a) is independent of the number ofgrid points, εKMC and εIMC increase linearly with N/Nmin.It follows that increasing the grid size requires larger val-ues of M in order to keep the sampling error small. How-ever, the grid error scales like Egrid ∝ 1/N2 = 2−2n forN ≥ Nmin [35]. We thus conclude that only moderateratios N/Nmin > 1 and therefore relatively small valuesof M are needed in order to achieve accurate solutionswith small grid errors.

The quantum ansatz in Fig. 1(b) is inspired from ten-sor network theory and can be regarded, for example, asthe Trotter decomposition of the time t evolution oper-ator exp(−iH(t)t/h) of a time-dependent spin Hamilto-nian H(t) with arbitrary short-range interactions actingon the initial state ∣0⟩ [67]. Similarly to the coupled clus-ter ansatz in quantum chemistry VQC calculations [19],there is currently no known efficient classical ansatz forthis state [68, 69]. From the VQC perspective, our quan-tum ansatz in Fig. 1(b) is composed of generic two-qubitgates that in an experiment are accurately approximatedby short sequences of gates if a sufficiently tunable oruniversal gate set is experimentally available [70]. Weenvisage that this quantum ansatz is more efficient than

4

10-2

10-1

100

0.02 0.04 0.06

32 64 1280

500

0.01 0.03 0.05

32 64 12802

10-2

100

0.0 0.5 1.010-1210-6100

FIG. 3. (a) Numerically exact solution ∣f(x)∣2 of Eq. (1)on a logarithmic scale, V (x) in Eq. (6) with s1 = 2 × 104 andκ1 = 2π × 32. The green thin line (blue normal line) is fors1/s2 = 200 (s1/s2 = 2). (b) Log-log plot of the IPR (toppanel) and lin-log plot of Smax of the exact solution ∣ψexact⟩(bottom panel) as a function of κ1. Green crosses (blue dots)correspond to s1/s2 = 200 (s1/s2 = 2). (c) Representationerror εR of the exact solution for the quantum ansatz (thicklines) and the MPS ansatz (thin lines) as a function of N /N .All curves are for s1/s2 = 2 and correspond to s1 = 2×104 andκ1 = 2π×32 (green dash-dotted), s1 = 8×104 and κ1 = 2π×64(blue dashed), and s1 = 3.2 × 105 and κ1 = 2π × 128 (purplesolid), respectively. The inset shows N as a function of κ1

for εR = 0.05 and s1/s2 = 2. Blue squares (green triangles)correspond to the quantum ansatz (MPS ansatz). All curvesin (a)-(c) are for N = 213 = 8192 grid points and g = 50.

methods based on an MPS ansatz on a classical computerlike the multigrid renormalization (MGR) method in [56].This MGR method is the most efficient and accurate clas-sical algorithm known to us for the problem consideredhere, for which it can already be exponentially faster thanstandard classical algorithms. A comparison with thispowerful classical method, which is based on variationalclassical MPS, will allow us to validate the superior vari-ational power of the quantum ansatz in Fig. 1(b) on aquantum computer. Note that the difficulty of solvinga classical optimization problem in many variables doesnot go away by using the quantum ansatz, as the actualoptimization is classical and there is no quantum advan-tage there. The quantum advantage stems solely fromthe faster evaluation of the cost function for our quan-tum ansatz in Fig. 1(b) on a quantum computer.

To provide numerical evidence for this we first ob-tain the numerically exact solution of Eq. (1) on theinterval [0,1] via the MGR algorithm [56] and by al-lowing for the maximal bond dimension χ of the MPS

ansatz [35]. In this case the numerically exact solutionis described by N = 2n parameters like in other conven-tional algorithms. The results are shown in Fig. 3 (a)for two different values of s1/s2. In the weakly disor-dered regime s1/s2 ≫ 1, ∣f(x)∣2 varies on the length scaleset by 1/κ1. On the contrary, the strongly disorderedregime s1/s2 ≈ 1 is characterized by strongly localizedsolutions in space. The localization of the wavefunctioncan be quantified using the inverse participation ratio(IPR) [71], IPR = (N ∑N−1

k=0 ∣ψk ∣4)−1. We show the IPR inFig. 3(b) as a function of κ1 (top panel) and find that itstays constant for s1/s2 ≫ 1. On the other hand, the IPRdecreases according to a power law with κ1 for s1/s2 ≈ 1,showing that the localized character of the wavefunctionincreases dramatically with κ1.

Next we encode the function values of the numericallyexact solution in the state ∣ψexact⟩ via Eq. (5) and calcu-late the maximum bipartite entanglement entropy Smax

of all possible bipartitions of the n-qubit wave function∣ψexact⟩. The quantity Smax is a measure of the entan-glement of ∣ψexact⟩ and is shown in the bottom panel ofFig. 3(b). The value of Smax is small and stays constantwith κ1 for s1/s2 ≫ 1 in the weakly disordered regime.Contrary to this, for s1/s2 ≈ 1 in the strongly disorderedregime we observe that Smax ∝ log(κ1).

The entanglement measure Smax provides a useful nec-essary criterion for efficient MPS approximations [72].A MPS of bond dimension χ can at most contain anamount of entanglement Smax[MPS] ≤ log(χ) [47, 50].For MPS to be efficient, we require that χ scales atmost polynomially with n, i.e. χ = poly(n), so thatSmax[MPS] ≤ log(poly(n)) and therefore MPS can onlycapture small amounts of entanglement efficiently. Thesmall values of Smax for s1/s2 ≫ 1 suggest that MPSwork well in the weakly disordered regime and indeedwe have confirmed numerically that this is true. There-fore, in the following we focus on the strongly disorderedregime s1/s2 ≈ 1. In this regime MPS cannot be an ef-ficient approximation for large values of κ1: The totalnumber of variational parameters of MPS N [MPS] de-pends quadratically on χ [35], i.e. N [MPS] ∝ χ2, and χneeds to grow polynomially with κ1, i.e. χ ∝ poly(κ1),to satisfy the observed entanglement scaling, such thatN [MPS] ∝ poly(κ1). Our quantum ansatz (QA) ofFig. 1(b) can capture much larger amounts of entan-glement Smax[QA] ≲ d efficiently [69] and therefore thisansatz can be an efficient approximation for large valuesof κ1: Because the total number of variational parame-ters N depends linearly on d for the quantum ansatz [35],i.e. N [QA]∝ d, and d just needs to grow logarithmically,i.e. d ∝ log(κ1), for the observed entanglement require-ments, we conclude that N [QA] ∝ log(κ1). These en-tanglement considerations show that the quantum ansatzhas the potential to be exponentially more efficient thanMPS in the strongly disordered regime for increasing val-ues of κ1.

To quantitatively analyze and demonstrate the effi-ciency of the quantum ansatz in this regime, we ob-

5

tain the set of parameters λ that maximize the fidelityF = ∣⟨ψexact∣ψ(λ)⟩∣ for different depths d [35]. The in-fidelity εR = 1 − F is thus a measure of the error whenapproximating the exact solution by this ansatz, and inthe following we refer to εR as the representation er-ror. As shown in Fig. 3(c), the representation errordecreases exponentially as a function of N for all val-ues of κ1 and therefore we obtain accurate solutions forN /N ≪ 1. Even for the largest value of κ1 = 2π × 128and εR ≈ 10−2, we find N /N ≈ 0.04 so that we onlyrequire 4% of the full number of parameters needed inconventional algorithms. Most importantly, the inset ofFig. 3(c) shows that, to obtain a fixed representationerror of εR = 0.05, the number of parameters of ourquantum ansatz needs to grow as N [QA] ∝ log(κ1).This numerical analysis therefore confirms our expec-tation, from the entanglement arguments in the previ-ous paragraph, that the quantum ansatz efficiently ap-proximates solutions in the strongly disordered regimeeven for large values of κ1. This is possible becausethe quantum ansatz captures the required entanglementSmax ∝ log(κ1) by means of just the small number ofparameters N [QA] ∝ log(κ1). The entanglement capa-bilities of MPS imply that N [MPS] ∝ poly(κ1) has tobe fulfilled. Therefore we conclude that our quantumansatz is exponentially more efficient than MPS in thestrongly disordered regime for growing values of κ1. Thiskey finding shows that nonlinear VQC can be exponen-tially more efficient than optimized, classical variationalschemes that are based on the MPS ansatz.

We now propose a calculation that illustrates the expo-nential advantage of our quantum ansatz on a quantumcomputer particularly clearly. We consider the groundstate problem of Eq. (1) with the quasi-random externalpotential of Eq. (6) on the interval [0,1] discretized byN = 2n equidistant grid points and use the FDM fromabove that leads to Eqs. (4a), (4b), and (4c). We choose

κ1 = 2π × 2n/2 and assume that n = 4,6,8, . . . qubitsstore the variational quantum ansatz. The smallest wavelength present in our external potential of Eq. (6) is deter-

mined by κ1 and reads 2π/κ1 = 2−n/2. The interval [0,1]accomodates 1/2−n/2 = 2n/2 of such wave lengths and ourgrid of N = 2n equidistant points resolves each such wavelength using N/2n/2 = 2n/2 grid points. Therefore therandomness of our quasi-random external potential ofEq. (6) as well as its FDM resolution grow exponentiallywith the number of qubits n. At the same time, all FDMerrors (using a FDM representation of the Laplace op-erator, the trapezoidal rule for integration, and so on)decrease polynomially with the grid spacing 1/N = 2−nand thus exponentially with n. Therefore, with grow-ing values of n, our FDM representation of Eq. (1) con-verges exponentially fast to the continuous problem and

the accuracy of randomness in Eq. (6) grows exponen-tially. Our proposal is to solve this problem for thestrongly disordered regime s1/s2 ≈ 1 in Eq. (6), i.e. com-pute the corresponding ground states, using an increasingnumber n = 4,6,8, . . .. Based on the numerical analy-sis of Fig. 3, we conjecture that MPS require resourcesN [MPS] ∝ poly(κ1) = poly(2π × 2n/2) = exp(n) grow-ing exponentially with n, whereas the quantum ansatzof Fig. 1(b) just needs resources N [QA] ∝ log(κ1) =log(2π × 2n/2) = poly(n) increasing polynomially with n.This calculation can be used to demonstrate quantumsupremacy as, by successively increasing n, the limit ofwhat is computationally possible on a classical computer(using MPS or exact classical methods) is reached quicklyand our quantum ansatz on a quantum computer remainsefficient far beyond this classical limit.

The quantum hardware requirements for this quantumsupremacy calculation go beyond the current capabilitiesof available NISQ devices [73]. Nevertheless the superiorperformance of our quantum ansatz is relevant for currentNISQ devices as only the most efficient variational statescan succeed in the presence of the current experimen-tal errors. We test the feasibility of nonlinear VQC onNISQ devices by calculating the ground state of Eq. (1)for a simple harmonic potential and a single variationalparameter on an IBM Q device [74] utilizing further net-work optimizations (see [35] for details). The experimen-tal implementation of the nonlinear VQC algorithm wasable to identify the optimal variational parameter withan error of less than 10% leading to excellent agreementof the ground state solutions with exact numerical solu-tions [c.f. Figs. 1(c)(d)].

The methods presented here are readily modified totwo-and three-dimensional problems with an overheadscaling linearly in the number of dimensions, and canbe applied to a broad range of nonlinear terms and dif-ferential operators. An exciting prospect for future workwill be to utilize intermediate-scale quantum computersfor solving non-linear problems on grid sizes beyond thescope of conventional computers.

ACKNOWLEDGMENTS

ML and DJ are grateful for funding from the Net-worked Quantum Information Technologies Hub (NQIT)of the UK National Quantum Technology Programmeas well as from the EPSRC grant “Tensor Net-work Theory for strongly correlated quantum systems”(EP/K038311/1). We acknowledge support from the EP-SRC National Quantum Technology Hub in NetworkedQuantum Information Technology (EP/M013243/1).MK and DJ acknowledge financial support from the Na-tional Research Foundation and the Ministry of Educa-tion, Singapore.

6

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8

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[74] We acknowledge use of the IBM Q for this work. The

views expressed are those of the authors and do not re-flect the official policy or position of IBM or the IBM Qteam.

Supplemental Material:Variational quantum algorithms for nonlinear problems

Michael Lubasch1, Jaewoo Joo1, Pierre Moinier2, Martin Kiffner3,1, and Dieter Jaksch1,3

Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom1

BAE Systems, Computational Engineering, Buckingham House,FPC 267 PO Box 5, Filton, Bristol BS34 7QW, United Kingdom2 and

Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 1175433

First we provide the details for the IBM quantum com-puter experiments in Sec. I. Then Sec. II contains a de-scription of the adder circuit used in the main text forcomputing the kinetic energy. Section III presents howto do time evolution with the Burgers equation using thevariational framework. We explain the matrix productstate (MPS) ansatz in Sec. IV and derive its quantumcircuit representation. The Monte Carlo sampling erroris explained in Sec. V. In Sec. VI we describe the fidelityoptimization algorithm and provide the number of varia-tional parameters in the quantum and MPS ansatz. Weuse the same notation and definitions as in the main text.

I. IBM QUANTUM COMPUTATIONS

To run on IBM’s noisy intermediate-term quantumcomputers, we need to reduce the hardware requirementsand optimize the circuits shown in the main text. Theseoptimizations lead to deviations from the generic non-linear VQC circuit structure discussed in the main text.For computing the expectation value of an operator Othat is diagonal, O = DO, or can be diagonalized easily,i.e. O = DO = ∑k ok ∣k⟩⟨k∣, we use a quantum circuit thatdoes not require ancilla qubits. The expectation valueof such an operator reads ⟨ψ∣DO ∣ψ⟩ = ∑k ok⟨ψ∣k⟩⟨k∣ψ⟩ =∑k ∣⟨k∣ψ⟩∣2ok. This expectation value is obtained by mea-suring all qubits and computing the mean value after Mexperiments as ⟨ψ∣DO ∣ψ⟩ ≈ (1/M)∑Mm=1 ok(m) , as shownin Fig. S1 (a). The following hardware-reduced quan-tum algorithms were originally inspired by the standardmethods that were used in previous experiments for theexperimental computation of expectation values of quan-tum chemistry Hamiltonians, e.g. [1, 2].

Each measurement m gives a multi-index binary(k)(m)from which we obtain k(m). Then the expectation val-ues read ⟨ψ∣DV ∣ψ⟩ ≈ (1/M)∑Mm=1 Vk(m) and ⟨ψ∣D∣ψ∣2 ∣ψ⟩ ≈(1/M)∑Mm=1 ∣ψk(m) ∣2. We compute ∣ψk ∣2 for a particularvalue of k by counting how often that value of k appearsin M experiments and then dividing that number by M .

Figure S1 (b) shows the corresponding quantum cir-cuit for the Laplace operator ∆. The Laplace opera-tor is diagonalized by the Quantum Fourier Transform(QFT) and has eigenvalues ∆k = 2N2 (cos(2πk/N) − 1)where N = 2n for n qubits. Therefore ⟨ψ∣∆∣ψ⟩ =⟨ψ∣( ˆQFT)†D∆( ˆQFT)∣ψ⟩ = ∑k ∣ψk ∣2∆k where ψk is ob-

tained from the QFT of ∣ψ⟩, i.e. ( ˆQFT)∣ψ⟩. After we

FIG. S1. (a) Quantum circuits for expectation value com-

putation of the potential ⟨ψ∣DV ∣ψ⟩ = ∑k ∣ψk ∣2Vk or nonlinear

term ⟨ψ∣D∣ψ∣2 ∣ψ⟩ = ∑k ∣ψk ∣2∣ψk ∣2 in the nonlinear Schrodinger

equation, where ∣ψ⟩ = U(λ)∣0⟩. (b) Quantum circuit for ex-

pectation value computation of the Laplace operator ⟨ψ∣∆∣ψ⟩.(c) The quantum ansatz that we have used as variational state

on the IBM quantum computer. Here Ry(λ) = exp(−iλσy/2)is the standard y rotation gate. The quantum ansatz has justone variational parameter, i.e. λ = λ. In the IBM experimentwe are restricting λ to be a real parameter and therefore thequantum ansatz has only real entries. (d) Quantum circuit forevaluating specific or averaged function values R{ψk} where

R{⋅} denotes the real part. The gates Ry(θ) are individuallycontrolled rotations around the y-axis and depending on θallow reading out function values and averages of R{ψk} bymeasuring the ancilla qubit in the computational basis [3, 4].For example, individual function values can be measured ef-ficiently at an arbitrary grid point k by choosing rotationangles θ = π binary(k). Averages over the least significantqubits can be formed by choosing the corresponding rotationangles π/2 when reading the function on a coarse grid. Weobtain the imaginary part I{ψk} by including an additionalphase shift gate R−π/2 = diag(1, exp(−iπ/2)) directly beforethe second Hadamard gate. Function values of more generalfunctions F = ∏rj=1(Ojf (j)) are evaluated by combining thecircuit presented here with the QNPU from Fig. 1 in the maintext.

have applied the QFT to ∣ψ⟩ = U(λ)∣0⟩, we compute theexpectation value as in Fig. S1 (a).

To illustrate that this quantum algorithm works, wechose the smallest non-trivial system size of n = 2qubits and the variational ansatz of Fig. S1 (c) whichis parametrized by one real parameter λ. To make useof the best available hardware, we chose to work onthe 20 qubit IBM devices Tokyo and Poughkeepsie. Onthe Tokyo device, the best T1 is 148.5µs, the best T2is 78.4µs, the best two-qubit error rate is 1.47%, andthe best single-qubit error rate is 0.064% [5]. On the

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Poughkeepsie device, the best T1 is 123.3µs, the best T2is 123.6µs, the best two-qubit error rate is 1.11%, andthe best single-qubit error rate is 0.052% [5]. These ex-perimental errors are just a little bit too large for ourquantum algorithm presented in the main text but aregood enough for our hardware-efficient quantum algo-rithm presented here (see also [6]). We considered thenonlinear Schrodinger equation with a harmonic trap po-tential V (x) = 2000(x − 0.5)2 and x ∈ [0,1). Using thevariational ansatz, we evaluated the quantum circuits ofFig. S1 (a) and (b) for all values of λ ∈ [0,2π) in stepsof 0.1. For each value of λ we computed the average ofthe particular quantity of interest over 24000 experimen-tal shots. This allowed us to compute the cost function(total energy in the nonlinear Schrodinger equation) ofFig. 1 in the main text and determine its minimum aswell as the corresponding solution function. Fig. 1 in themain text shows the squared absolute value of the so-lution functions, which is also referred to as the groundstate solution (density). This is compared to the nu-merically exact solution obtained from imaginary timeevolution [7].

Figure S1 (d) shows the general quantum circuit forevaluating specific or averaged function values. We didnot use this circuit in the IBM experiment where, instead,we used the circuit of Fig. S1 (a) to compute ∣ψ0∣2, ∣ψ1∣2,∣ψ2∣2, and ∣ψ3∣2.

Note that the QFT required here could be realized us-ing just single-qubit gates with classical control [8], whichwould simplify the hardware requirements significantly.We did not exploit this simplification here since the IBMquantum computers do not offer classical control yet.Therefore we used the complete QFT quantum circuitshown in Fig. S1 (b).

II. ADDER CIRCUIT FOR KINETIC ENERGY

The quantum circuit presented in Fig. S2 is the QNPUfor computing the kinetic energy in the main text via theadder operator A. This operator increments the index ofevery wave function coefficient by one and can thereforebe seen as a special case of the general quantum circuitsfor arithmetic operations discussed in Refs. [9, 10]. Com-pared with the more general circuits of Refs. [9, 10], thisadder circuit requires fewer ancilla qubits and gates. Wededuce from Fig. S2 that for n > 2 qubits, the adder cir-cuit is composed of n− 2 ancilla qubits, n− 2 CNOT and2n − 2 Toffoli gates. For n = 2 the adder circuit requiresone CNOT and one Toffoli gate, and for n = 1 one CNOTgate suffices.

III. NONLINEAR VQC FOR THE BURGERSEQUATION

To illustrate how a time-dependent nonlinear partialdifferential equation is solved, we develop a quantum al-

FIG. S2. QNPU for computing the kinetic energy in themain text via the adder operator A. This operator actson a state ∣ψ⟩ = ∑N−1k=0 ψk ∣binary(k)⟩ in such a way that

A∑N−1k=0 ψk ∣binary(k)⟩ = ∑N−1k=0 ψk+1∣binary(k)⟩. We considerperiodic boundary conditions where ψN = ψ0. Inserting thisQNPU in Fig. 1 (a) in the main text gives the circuit for

determining the kinetic energy ⟪K⟫. Note that U(λ) is fed

on input port IP1 and in ∣ψ⟩ = U(λ)∣0⟩, the lowermost qubitis the most significant qubit and the uppermost qubit is theleast significant.

gorithm that approximates time evolution of the Burgersequation

∂

∂tf(x, t) = ν ∂2

∂x2f(x, t) − f(x, t) ∂

∂xf(x, t) . (S1)

Here, ν is the so-called coefficient of kinematic viscos-ity and Eq. (S1) gives rise to turbulence when 1/ν be-comes large [11, 12]. We discretize the spatial coordinatex (as done in the main text of this article for the nonlin-ear Schrodinger equation) and we represent the resultingfunction values fk at time t by a vector ∣f⟩. Then Eq. (S1)takes on the form:

∂

∂t∣f⟩ = ν∆∣f⟩ −Df∇∣f⟩ , (S2)

where ∆ and ∇ are the discretized Laplace and Nablamatrix, respectively, and Df is a diagonal matrix withthe values fk on its diagonal.

For the classical simulation of Eq. (S2), we usethe time-dependent variational principle algorithm forMPS [13, 14]. This algorithm requires that ∆ and ∣f⟩∇are written as matrix product operators and we show howthis is done in Ref. [15].

For the quantum algorithm, it is important to notethat Eq. (S2) does not conserve the norm of ∣f⟩. There-fore we choose the variational ansatz to be ∣f⟩ =λ0∣ψ(λ)⟩, where λ0 is a new variational parameter andλ = (λ1, λ2, . . .) as before (see discussion in main text).The introduction of λ0 allows us to handle arbitrarynorms ⟨f ∣f⟩ of the variational solution ∣f⟩, while thewave function on the quantum computer always fulfills⟨ψ(λ)∣ψ(λ)⟩ = 1.

To keep this discussion as simple as possible, we justconsider the Euler method for time evolution [16] here.

S3

The Euler method computes the time-evolved solution af-ter time step τ , ∣f(t+τ)⟩, by using the previous solution,∣f(t)⟩, and the right-hand side of Eq. S2 at time t. If wesummarize the right-hand side of Eq. S2 by O∣f⟩, thenthe Euler method identifies ∣f(t+τ)⟩ = (1 + τO(t)) ∣f(t)⟩.In the variational setting, instead of directly computing∣f(t + τ)⟩ via this formula, it is more efficient to definethe cost function

C(∣f(t + τ)⟩) = ∣∣∣f(t + τ)⟩ − (1 + τO(t))∣f(t)⟩∣∣2 (S3)

and minimize this cost function via the variational pa-rameters of ∣f(t + τ)⟩.

For the quantum algorithm we define ∣f(t + τ)⟩ =λ0∣ψ(λ)⟩ = λ0U(λ)∣0⟩ and ∣f(t)⟩ = λ0∣ψ⟩ = λ0

ˆU ∣0⟩ forevery value of t. Then for each time step τ , the followingcost function needs to be minimized:

C(λ0,λ) = ∣∣λ0∣ψ(λ)⟩ − (1 + τO)λ0∣ψ⟩∣∣2 (S4)

= ∣λ0∣2 − 2R{λ0λ∗0⟨ψ∣(1 + τO)∣ψ(λ)⟩} + const.

= ∣λ0∣2 − 2R{λ0λ∗0⟨0∣ ˆU †(1 + τO)U(λ)∣0⟩} + const.,

where O = ν∆ − λ0Dψ∇. For each time step, Eq. (S4) is

minimized via the procedures discussed in Refs. [2, 17].Figure S3 shows the quantum circuits that are requiredfor the computation of this cost function. Note that the

QNPUs need to be modified after every time step, as ˆUchanges after every time step.

More sophisticated time evolution algorithms can read-ily be realized for nonlinear partial differential equa-tions on a quantum computer. For example, a time-dependent variational principle can be formulated simi-larly to Refs. [18, 19] where the required new ingredients(for nonlinear terms) are obtained from the quantum cir-cuits presented here and in the main text.

IV. MPS ANSATZ

In this section, we describe the general procedure fortransforming MPS into quantum circuits. We also dis-cuss how plane wave, cosine, and sine functions are rep-resented on a quantum computer.

A. General Procedure

An MPS of bond dimension χ,

∣ψMPS⟩ = ∑q1,...,qn

B[1]q1 . . .B[n]qn ∣q1, . . . , qn⟩ , (S5)

is defined in terms of n tensors B[j]qjαj−1,αj , where theindex qj ∈ {0,1} and all indices αj run from 1 to χ ex-cept α0 and αn which both take on only the value 1. TheMPS tensor entries are real or complex numbers and con-stitute the variational parameters in the MPS ansatz. We

FIG. S3. QNPUs for evaluating the cost function Eq. (S4)of the Burgers equation for an example with n = 4. TheseQNPUs need to be inserted in the circuit of Fig. 1 (a) fromthe main text with a small modification: in this circuit theunitary on input port IP1 needs to be controlled by the an-cilla (as are all unitaries on the other input ports IP2 to IPr).

(a) ⟨ψ∣ψ⟩ = ⟨0∣ ˆU†U(λ)∣0⟩. (b) ⟨ψ∣A∣ψ⟩ = ⟨0∣ ˆU†AU(λ)∣0⟩. (c)

⟨ψ∣AD†ψ∣ψ⟩ = ⟨0∣ ˆU†AD†

ψU(λ)∣0⟩. Here ˆU† is the adjoint of ˆU

which defines ∣ψ⟩ = ˆU ∣0⟩, i.e. the function ∣f(t)⟩ = λ0∣ψ⟩ at

time t. Note that ˆU† is fixed and has no variational parame-ters. The variational parameters reside outside the QNPUs inU(λ) which defines ∣f(t + τ)⟩ = λ0∣ψ⟩ = λ0U(λ)∣0⟩. Further-

more, in the notation here, A is the adder circuit defined inthe caption of Fig. S2, and Dψ denotes the diagonal operator

that has the values of ψk on its diagonal.

observe that there are O(nχ2) = O(poly(n)) variationalparameters, bipartite entanglement entropy is limited tovalues ≲ log(χ), and most MPS algorithms have a com-putational cost O(poly(χ)) [20, 21].

Figure S4 shows the required steps for transforming aMPS of bond dimension χ into a quantum circuit. TheMPS quantum circuit consists of n − s unitary matriceseach of dimension 2s+1×2s+1 = 2χ×2χ. Each of these uni-tary matrices can be further decomposed into a productof O(χ2) generic two-qubit unitaries [22]. We can em-bed these resulting two-qubit unitaries in the quantumansatz of Fig. 1 (b) in the main text. Therefore any MPSof bond dimension χ ∼ poly(n) is efficiently contained inthe quantum ansatz of depth d ∼ poly(n). Figure S5 il-lustrates how the resulting quantum circuits look like foran MPS of bond dimensions χ = 2 and 4. For the depthsd of these quantum circuits we obtain d = n − 1 for χ = 2and d = 5(n − 2) + 1 for χ = 4.

B. Plane Wave, Sine, and Cosine FunctionRepresentations on a Quantum Computer

MPS contain polynomials as well as Fourier series [24,25] and so these are also contained in the quantum ansatz

S4

FIG. S4. Transformation from MPS into quantum circuit.(a) We first compute the value s = ⌈log2(χ)⌉, where ⌈⋅⌉ is theceiling function. This allows us to rewrite the MPS with bonddimension χ = 2s. If χ > χ then we achieve this by embed-ding each tensor B[j] in a new larger tensor B[j] where theadditional new elements are set to zero. (b) Then successiveQR decompositions turn this MPS into a product of isometricmatrices (and possibly a normalization factor). We start with

the QR decomposition of B[1] and move towards and end

with B[n]. After the QR decomposition of B[j] we multiply

R[j] with B[j + 1] before performing the QR decomposition

of B[j+1]. The resulting tensor network consists of isometricmatrices only. (c) We multiply all tensors left of Q[s+1] with

Q[s + 1] to obtain Q[s + 1]. The resulting isometric matricesare then embedded in unitary matrices U[j]. Rearranging thefinal network gives the MPS quantum circuit.

presented in Fig. 1 (b) in the main text. A plane wavefunction exp(iκx) can be written as an MPS of bonddimension χ = 1 [24] and its quantum circuit represen-tation requires only single-qubit gates, as can be seen inFig. S6 (a). We write the function sin(κx) as an MPSof bond dimension χ = 2 [24], which is a quantum circuitof the form shown in Fig. S5 (a). We write the sum oftwo different sine functions as an MPS of bond dimensionχ = 4, which is a quantum circuit of the form shown inFig. S5 (b) and (c). This MPS of bond dimension χ = 4is used in the main text for representing the potentialV (x) = s1 sin(κ1x) + s2 sin(κ2x).

Depending on the hardware capabilities, there existseveral alternatives for realizing V (x) = s1 sin(κ1x) +s2 sin(κ2x) on a quantum computer. Note that sin(κx)

FIG. S5. Example MPS quantum circuits. (a) MPS of bonddimension χ = 2 represented in terms of n − 1 generic two-qubit unitaries. (b) MPS of bond dimension χ = 4 representedin terms of n − 2 generic three-qubit unitaries. (c) MPS ofbond dimension χ = 4 represented in terms of 5(n − 2) + 1generic two-qubit unitaries using the decomposition of three-qubit unitaries of Ref. [23].

FIG. S6. (a) This quantum circuit realizes the wave function∣ψ⟩ = (1/√2n)∑q1,q2,...,qn exp(iκ∑nj=1 qj2−j)∣q1, q2, . . . , qn⟩which represents a plane wave of wave vector κ.Rϕ = diag(1, exp(iϕ)) is a phase shift gate of phaseshift ϕ. (b) Quantum circuit for the probabilistic generationof ψ(x) ∝ cos(κx) and ψ(x) ∝ sin(κx) with probability0.5, respectively. When 0 is measured in the uppermostqubit, then ∣ψ⟩ = (1/√2n)∑q1,q2,...,qn exp(iκ∑nj=1 qj2−j) +exp(−iκ∑nj=1 qj2−j)∣q1, q2, . . . , qn⟩, and the wave func-tion coefficients are proportional to the cosine func-tion. When 1 is measured in the uppermost qubit,then ∣ψ⟩ = (1/√2n)∑q1,q2,...,qn exp(iκ∑nj=1 qj2−j) −exp(−iκ∑nj=1 qj2−j)∣q1, q2, . . . , qn⟩, and the wave functioncoefficients are proportional to the sine function.

as well as cos(κx) can be written as a sum of two planewaves, respectively. Therefore two quantum circuits ofthe form of Fig. S6 (a) suffice to represent a single func-tion sin(κx). Four quantum circuits of the form ofFig. S6 (a) represent the potential V (x) = s1 sin(κ1x) +s2 sin(κ2x). Therefore, instead of a single MPS quantumcircuit, we can also use several simpler circuits (composedof single-qubit unitaries only) for the potential represen-tation on a quantum computer. Sine and cosine functionsare also generated probabilistically with the quantum cir-cuit of Fig. S6 (b).

S5

V. SAMPLING ERROR

Here we analyze the grid error and the Monte Carlosampling error. We first investigate the convergence ofthe numerically exact solution with the number of gridpoints. These results facilitate the analysis of the MonteCarlo sampling error.

A. Grid Error

We numerically calculate the expectation values ⟪K⟫,⟪P⟫ and ⟪I⟫ in the main text for different grid sizesN = 2n. The results are presented in Fig. S7 and showthat all expectation values increase with an increasingnumber of grid points until they level off and converge ata critical number of grid points Nmin = 2nmin . This crit-ical number depends only on the value of the wavenum-ber κ1 which controls the shape of the external potential.More specifically, Nmin approximately coincides with thenumber of grid points required to resolve the potential

V (x) = s1 sin(κ1x) + s2 sin(κ2x) . (S6)

By taking into account that one needs at least four gridpoints to resolve a single wave length, we obtain nmin = 6for κ1 = 2π × 16, nmin = 7 for κ1 = 2π × 32, nmin = 8 forκ1 = 2π×64, and nmin = 9 for κ1 = 2π×128. These simpleestimates are in very good agreement with the results inFig. S7. More generally, we define

Nmin = `

`min, (S7)

where `min is the smallest lengthscale of the problem.The expectation values ⟪⋅⟫ converge if the grid size obeysN > Nmin, and only in this regime the grid error scaleslike Egrid ∝ 1/N2.

The expectation values ⟪X⟫ are related in the maintext to expectation values of an ancilla qubit ⟨σz⟩Xanc

when measured on a quantum computer. The resultsfor ⟨σz⟩Xanc are also shown in Fig. S7 and illustrate howthe values ⟨σz⟩Xanc converge for N > Nmin. We observethat ⟨σz⟩Panc converges to a constant value and

1 − ⟨σz⟩Kanc ∝ 2−2n , (S8a)

⟨σz⟩Ianc ∝ 2−n , (S8b)

for n > nmin. In particular, we thus conclude ⟨σz⟩Kanc → 1and ⟨σz⟩Ianc → 0 for n → ∞. Note that the expectedscaling in Eq. (S8) can be used in practice to determinethe minimal grid size Nmin. This is important for findingthe optimal grid size that results in converged resultswith reasonable Monte Carlo sampling sizes, as discussedin the next section.

B. Monte Carlo Sampling Error

The expectation values ⟨σz⟩anc in Eqs. (6) and (7) ofthe main text represent exact quantum mechanical ex-

pectation values. Measuring these quantities on a quan-tum computer involves averaging over M Monte Carlosamples,

⟨σz⟩anc = 1

M

M∑m=1⟨σz⟩m , (S9)

where ⟨σz⟩m denotes the eigenvalue of σz obtained forthe mth sample. In general, the Monte Carlo samplingerror EMC = ⟨σz⟩anc − ⟨σz⟩anc associated with drawing Mrandom samples is [16]

EMC =√⟨σ2

z⟩anc − ⟨σz⟩2anc√M

, (S10a)

=√

1 − ⟨σz⟩2anc√M

, (S10b)

where we used σ2z = 1 in the second line.

With the help of these definitions we obtain

EPMC = α√

1 − [⟨σz⟩Panc]2√M

, (S11a)

EKMC = N2

`2

√1 − [⟨σz⟩Kanc]2√

M, (S11b)

EIMC = 1

2gN

`

√1 − [⟨σz⟩Ianc]2√

M, (S11c)

where EXMC is the absolute Monte Carlo error associatedwith the average ⟪X⟫. The expressions in Eq. (S11) canbe further simplified, and we begin with the error as-sociated with the potential energy ⟪P⟫. With the helpof Eq. (7)(a) in the main text, the relative Monte Carlosampling error can be written as

εPMC = EPMC⟪P⟫ = CP 1√M

, (S12)

where

CP =¿ÁÁÀ α2

⟪P⟫2− 1 . (S13)

Since α2/⟪P⟫2 ≥ 1 by virtue of the definition of α inthe main text, CP is always real. For the parameters ofFig. S7 CP takes on values between 64.3 and 67.2.

Next we investigate the error associated with the ki-netic energy term ⟪K⟫ and note that

1 − [⟨σz⟩Kanc]2 = (1 − ⟨σz⟩Kanc) (1 + ⟨σz⟩Kanc) , (S14a)

1 − ⟨σz⟩Kanc = h2N⟪K⟫ . (S14b)

Combining the equations above we obtain the relativesampling error associated with ⟪K⟫,

εKMC = EKMC⟪K⟫ = N`

√1 + ⟨σz⟩Kanc√⟪K⟫

1√M

. (S15)

S6

4 6 8 10 12

105

104

103

102

4 6 8 10 12

104

103

102

4 6 8 10 12

10-1

10-410-310-2

100

4 6 8 10 12

10-1

10-310-2

100

4 6 8 10 12

105

104

103

10-1

10-2

4 6 8 10 12

FIG. S7. Energy expectation values ⟪K⟫ = 4nR{⟨ψ∣(1 −A)∣ψ⟩} (a), ⟪P⟫ = R{⟨ψ∣V ∣ψ⟩} (b), ⟪I⟫ = g2nR{⟨ψ∣D∣ψ∣2 ∣ψ⟩}(c), and the corresponding values related to the ancilla qubitmeasurement: 1− ⟨σz⟩Kanc = ⟪K⟫/4n (d), ⟨σz⟩Panc = ⟪P⟫/α (e),

and ⟨σz⟩Ianc = ⟪I⟫/(g2n) (f). Here A is the adder operator

defined in the caption of Fig. S2, V represents the potentialV (x) = s1 sin(κ1x) + s2 sin(κ2x), and D∣ψ∣2 is diagonal with

the values of ∣ψk ∣2 on its diagonal. We consider the stronglydisordered regime of Fig. 3 in the main text (where s1/s2 = 2

and κ2 = 2κ1/(1+√5)) with s1 = 5×103 and κ1 = 2π×16 (red

dotted), s1 = 2 × 104 and κ1 = 2π × 32 (green dash-dotted),s1 = 8 × 104 and κ1 = 2π × 64 (blue dashed), s1 = 3.2 × 105

and κ1 = 2π ×128 (purple solid), and we compare g = 10 (thinlines) with 50 (normal lines). All results are for N = 213 gridpoints.

To further simplify this we assume that N ≥ Nmin suchthat ⟨σz⟩Kanc ≈ 1 [see Eq. (S8)] and utilize Eq. (S7),

εKMC ≈ CK N

Nmin

1√M

, (S16)

where

CK = √2

lmin

√⟪K⟫ . (S17)

The constant CK is expected to be of the order of unitysince ⟪K⟫ ≈ 1/`2min. For the parameters of Fig. S7 CKtakes on values between 2.24 and 2.30.

Finally, we consider the nonlinear term and point outthat ⟨σz⟩Ianc → 0 for N → ∞, see Eq. (S8). The leading

term in Eq. (S11c) is thus EIMC ≈ gN/(2`√M), and therelative Monte Carlo sampling error is

εIMC = EIMC⟪I⟫ ≈ CI N

Nmin

1√M

, (S18)

where

CI ≈ gNmin

2`⟪I⟫ . (S19)

For the parameters of Fig. S7 CI takes on values between2.58 and 6.88.

The preceeding analysis concludes the proof of Eq. (8)in the main text. For N < Nmin all observables require ofthe order of M ≈ ε−2

MC Monte Carlo samples in order tobe accurately measured. The same result holds for thepotential term even if N ≥ Nmin. On the other hand,⟪K⟫ and ⟪I⟫ require M ≈ 4nε−2

MC experiments and thusM increases exponentially with n. The grid error alsotends to zero exponentially fast ∝ 4−n for n > nmin, andthus n only needs to be slightly larger than nmin in orderto achieve accurate solutions. This avoids exponentiallymany Monte Carlo samples.

Note that the minimal grid size Nmin can be deter-mined by gradually increasing N until the measured ex-pectation values ⟨σz⟩anc show the expected scaling be-haviour in Eq. (S8). In this way optimal grid sizes re-sulting in accurate solutions with moderate Monte Carlosampling sizes can be chosen.

VI. FIDELITY F OPTIMIZATION ANDNUMBER OF VARIATIONAL PARAMETERS N

In the main text of this article we determine opti-mal approximations in terms of MPS ∣ψMPS⟩ and interms of our quantum ansatz ∣ψQA⟩ for numerically ex-act solutions ∣ψexact⟩. We achieve this by maximizingF = ⟨ψexact∣ψMPS⟩ as well as F = ⟨ψexact∣ψQA⟩ usingstandard tensor network optimization techniques [20, 21].Our algorithm optimizes the variational unitaries one af-ter another and for each unitary Uj determines the new

unitary Uj that maximizes F under the assumption thatall other unitaries are fixed. Because F = tr(UjRj) we

obtain the optimal Uj from a singular value decomposi-

tion of Rj = URjΣRjVRj as Uj = V †RjU †Rj

.

For real-valued unitary gates (as considered through-out our analysis), our quantum ansatz of Fig. 1 (b) inthe main text has 3 variational parameters per two-qubitunitary in the first column and then 6 variational param-eters per two-qubit unitary in all the other columns. AMPS tensor B[j]qjαj−1,αj has dim(αj−1)dim(αj)dim(qj)−dim(αj)(dim(αj) + 1)/2 variational parameters. Thisis derived by reshaping B[j]qjαj−1,αj = B[j](αj−1,qj),αj

asan isometric matrix with row-muliindex (αj−1, qj) andcolumn-index αj . This reshaped form allows us to countthe number of free parameters of such an isometry.

S7

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