Validating a Two Qubit Non-Stoquastic Hamiltonian in Quantum Annealing

Tameem Albash1 1Department of Electrical and Computer Engineering, Department of Physics and Astronomy, and Center for Quantum Information and Control, CQuIC, University of New Mexico, Albuquerque, New Mexico 87131, USA We propose a two qubit experiment for validating tunable antiferromagnetic XX interactions in quantum annealing. Such interactions allow the time-dependent Hamiltonian to be non-stoquastic, and the instantaneous ground state can have negative amplitudes in the computational basis. Our construction relies on how the degeneracy of the Ising Hamiltonian’s ground states is broken away from the end point of the anneal: above a certain value of the antiferromagnetic XX interaction strength, the perturbative ground state at the end of the anneal changes from a symmetric to an antisymmetric state. This change is associated with a suppression of one of the Ising ground states, which can then be detected using solely computational basis measurements. We show that a semiclassical approximation of the annealing protocol fails to reproduce this feature, making it a candidate ‘quantum signature’ of the evolution.

I. INTRODUCTION the strength of the antiferromagnetic XX interaction is increased, the ground state changes from a symmetric combination of the three ground states of H to an anti- There remain no demonstrated examples of a quan- P symmetric combination of two of the three ground states tum speedup using quantum annealing [1–5] outside of of H . The suppression of population in one of the the oracular setting [6,7]. Evidence that is often cited P ground states is then a measurable signature of this tran- for this predicament is that standard quantum annealing sition. The proposal thus tests both the tunability of the implements a stoquastic Hamiltonian [8–13] throughout interactions and the ability of the quantum annealer to the anneal, and ground state adiabatic quantum comput- implement a ground state with non-trivial relative phases ing [14] with a stoquastic Hamiltonian is not expected to between the computational basis states. We demonstrate be universal [9]. The restriction to stoquastic Hamilto- that at least one semiclassical model of quantum anneal- nians, for which a classical probability distribution can ing, whereby qubits are replaced by spin vectors, fails to be associated with the ground state, also makes the an- reproduce this signature. nealing protocol amenable to classical simulation using Our paper is structured as follows. In Sec.II, we show Quantum Monte Carlo (QMC) techniques. While QMC how perturbation theory predicts the breaking of the de- does not simulate the unitary dynamics of quantum an- generacy of the final three ground states to give a unique nealing, it does reproduce the scaling dependence with ground state near the end of the anneal. In Sec.III, the minimum gap for certain problems [15], but crucially we present the results of dynamical simulations to test it fails to do so for others [16, 17], making it unclear a the robustness of the proposal to several noise models. priori for which classes of problems quantum annealing In Sec.IV, we provide a comparison to a semiclassical can provide a legitimate scaling advantage over QMC. model of quantum annealing, and we conclude in Sec.V. The introduction of novel interactions at intermedi- ate points in the anneal that make the Hamiltonian non- stoquastic would hinder QMC techniques from being ef- II. PERTURBATION THEORY ficient simulators of the annealing protocol because of the associated sign-problem [11–13, 18]. This effectively We begin by considering the following time-dependent eliminates QMC as a legitimate classical competitor, and 2-qubit Hamiltonian:

arXiv:1909.06333v2 [quant-ph] 9 Jan 2020 it remains an open question to what extent this will im- x x x x prove the current situation of demonstrating a quantum H(s)/(~ω) = −(1 − s)(σ1 + σ2 ) + αs(1 − s)σ1 σ2 z z z z speedup using quantum annealing [19–23][24]. +s (−σ1 − σ2 + σ1 σ2 ) , (1) Nevertheless, experimental realizations of such interac- γ where σi is the Pauli-γ operator acting on qubit i, |α| tions are ongoing [25, 26], and here we propose a method is the strength of the XX interaction with α < 0 or > to validate the implementation of tunable antiferromag- 0 corresponding to a ferromagnetic or antiferromagnetic netic XX interactions within the constraints of the quan- interaction, and ~ω sets the overall energy scale. Here HP tum annealing protocol, i.e. the evolution terminates on is given by the last term in parenthesis. The Hamiltonian a Hamiltonian HP that is diagonal in the computational is invariant under the interchange of the two qubits, so all basis, and only measurements in the computational basis energy eigenstates will be invariant up to an overall phase at the end of the anneal are allowed. under the nterchange of the two qubits. The ground state In order to accomplish this objective, our construc- of the Ising Hamiltonian at s = 1 exhibits a three-fold tion relies on a change in the ground state of the time- degeneracy: |00i, |01i, |10i, where |0i is the eigenstate of dependent Hamiltonian near the end of the anneal. As σz with positive eigenvalue. 2

If we denote the perturbation parameter by Γ = 1 − s, III. DYNAMICS expanding H(s) around s = 1 gives to first order H(s) = H(1) + ~ωΓV1, where Because the energy level crossing discussed above is associated with a symmetry of the Hamiltonian, an adi- x x x x z z z z V1 = −(σ1 + σ2 ) + ασ1 σ2 − (−σ1 − σ2 + σ1 σ2 ) , (2) abatic evolution will not follow the global ground state through the crossing. Therefore, we propose to break When V1 is projected onto the ground state subspace the qubit permutation symmetry by offsetting one of the of s = 1, the three eigenstates with their corresponding transverse fields: eigenvalues of the resulting operator are given by: H(s) x x x x = −(1 − s)(σ1 + (1 − β)σ2 ) + αs(1 − s)σ1 σ2 1 ~ω |λi = √ (|01i − |10i) , λ = 1 − α , (3a) +s (−σz − σz + σzσz) , (5) 2 1 2 1 2 0 1 0 with β > 0. The true level-crossing associated with |λ i = q (γ+|00i + |01i + |10i) , λ = 1 + γ−, γ2 + 2 α > 1 now becomes an avoided level crossing, and an + adiabatic evolution is able to follow the global ground (3b) state throughout the anneal. The change in the charac- 1 ter of the ground state remains apparent for a range of β |λ00i = (γ |00i + |01i + |10i) , λ00 = 1 + γ , q − + values as α is tuned, as we show in Fig.1. The transition γ2 + 2 − at α = 1 becomes sharper as β → 0. (3c) We now consider the robustness of our proposal to sev- √ eral noise models. We first consider the effect of inter- 1 2
0 00 actions with a thermal environment as described by a with γ± = 2 α ± 8 + α . The states |λ i and |λ i are symmetric under the interchange of the qubits, whereas weak-coupling limit master equation [27]. We give de- |λi is antisymmetric under the interchange of the qubits tails of this master equation in AppendixA. We show in and does not have any weight on the |00i state. A clas- Fig. 2(a) the behavior as a function of the dimensionless sical thermal state at s = 1 will by definition have equal system-bath coupling κ2. We see that for α = 2, the populations on all three ground states, approaching the closed system evolution (κ2 = 0) is close to adiabatic for 3 value 1/3 in the limit of zero temperature. ωtf & 5×10 . As soon as we have a non-zero system bath The state with smallest eigenvalue {λ, λ0} is the ground interaction, the population of the |00i state approaches state for Γ = 0+, so for α < 1 the ground state is |λ0i and the classical Ising Gibbs state population of 1/3 in the for α > 1, the ground state is |λi. As α is swept through long time limit, as expected by the open-system adiabatic 2 this point, the ground state changes from a symmetric to theorem [28]. If κ is sufficiently small, we observe the an antisymmetric state. Quantitatively, we can consider expected competition between closed and open system the expectation value of the SWAP operator, SWAP = adiabaticity: closed-system adiabaticity sends the pop-   ulation in the |00i state to zero, whereas open-system 1 11 + P σγ σγ , which changes from +1 to −1 2 γ={x,y,z} 1 2 adiabaticity sends the population to 1/3. For sufficiently as we cross α = 1. small κ2 the closed system adiabatic time scale is smaller Because the ground state at s = 0 is symmetric, for than the open system time scale, and we can continue α > 1 there is a true level-crossing in the spectrum as- to observe our desired signature of population suppres- sociated with this change in ground state. This level sion of the |00i state before open-system effects begin to crossing is predicted already by considering the perturba- dominate. tive corrections to the energies associated with the states We also consider the effect of implementation errors 0 |λi and |λ i at second-order in perturbation theory. The in the parameters of HP to our population suppression non-zero contributions are given by: signature. We model these as independent Gaussian ran- dom variables with zero mean and standard deviation σ. E(2)(Γ) 2 In Fig. 2(b), we show the populations in the computa- = − 1 + hλ|V1|λiΓ + hλ|V2|λiΓ , (4a) ~ω tional basis states as a function of σ, where we again E0 (Γ) observe that there is a noise threshold before the distri- (2) 0 0 = − 1 + hλ |V1|λ iΓ bution looks classical. We emphasize that the evolution ~ω   for low σ is non-adiabatic, even though we picked a time- 0 0 1 0 2 2 + hλ |V2|λ i − |hλ |V1|11i| Γ , (4b) scale that is close to adiabatic for the original Hamil- 4 tonian. This is critical since a random noise instantia- tion picks one of the three ground states to be the new x x where V2 = −ασ1 σ2 is a second-order perturbation that ground state with equal probability, so in the adiabatic also contributes. For α > 0, E(2) curves upwards and limit of the noisy Hamiltonian, we recover the classical 0 E(2) curves downwards, but only for α > 1 does E(2) result upon averaging over noise realizations. Therefore, 0 reach a lower value than E(2) for s < 1, which leads to we find that we can continue to observe our desired sig- the two energies crossing at some intermediate s value. nature of population suppression if there is a separation 3 between the noise-less and noisy adiabatic time scales. where ~ x z H1 = 2 (−(1 − s) + αs(1 − s)M2 )x ˆ + 2s (−1 + M2 )z ˆ , 1 0.8 (7a) 0.5 0.6 ~ x H2 = 2 (−(1 − s)(1 − β) + αs(1 − s)M1 )x ˆ 0.4 z 0 + 2s (−1 + M1 )z ˆ . (7b) -0.5 0.2 (For a detailed derivation, see Ref. [31, 32].) We show

SWAP expect. value -1 0 in Fig. 3(a) the resulting magnetization at s = 1 for -2 -1 0 1 2 -2 -1 0 1 2 increasing ωtf , where we see the system approaches the ~ (a) (b) values θ = (0, π/2) at very long times. In order to understand the long time behavior, it is use- FIG. 1. Properties of the perturbative ground state calculated ful to consider the potential energy landscape on which from first order perturbation theory for various β values. (a) the dynamics occurs: Expectation value of the SWAP operator. (b) Population in V (s) = −(1 − s)(M x + (1 − β)M x) + αs(1 − s)M xM x the state |00i. 1 2 1 2 z z z z +s (−M1 − M2 + M1 M2 ) , (8) which amounts to replacing the Pauli operators in Eq. (5) by σx 7→ M x, σz 7→ M z. For β = 0 and φ = 0, the two 0.4 0.6 i i i i i dimensional potential energy landscape as a function of 0.3 ~ 0.4 θ exhibits the features we associate with second-order 0.2 phase transitions as we increase s. Specifically, a single 0.2 global minimum bifurcates into two equal in energy min- 0.1 Populations ima, which move towards the positions θ~ = (π/2, 0) and 0 0 (0, π/2) respectively as s goes to 1, irrespective of the 0 1 2 10 -3 10 -2 10 -1 10 4 value of α. For finite β, one of the two minima is always energetically favored (shown in Fig. 3(b)), and for a suffi- (a) (b) ciently slow evolution, the system follows this minimum. FIG. 2. (a) Population in the |00i state at the end of the anneal for a system evolving according to the weak-coupling 1 master equation simulations with α = 2, β = 0.05 , dimension- 2 -0.2 less system-bath coupling κ , energy scale kB T/~ω = 1.57, 0.5 and varying total annealing time tf . (b) Computational ba- -0.4 sis populations of the evolved state at the end of the anneal 0 -0.6 4 2 for β = 0.05, ωtf = 10 , κ = 0,α = 2 with Gaussian noise -0.8 N ∼ (0, σ2) on the Ising parameters. Results are for the mean Magnetization 3 -0.5 -1 value of 10 independent noise realization, and the error bars 10 -2 10 0 10 2 10 4 correspond to the 95% confidence interval calculated using a bootstrap. (a) (b)

FIG. 3. (a) Spin-vector dynamics results for α = 2 and β = 0.05. (b) Cut through the semiclassical potential for fixed φi = 0 for the system parameters α = 2, β = 0.05, s = 0.35.

IV. SEMICLASSICAL ANALOGUE We can also consider an evolution of the spin vectors We consider a semiclassical limit of qubits corre- described by Monte Carlo updates, whereby a random sponding to the saddle-point approximation of the M~ i is drawn, and the update is accepted according to path integral for the qubit system [29], where the the Metropolis-Hastings criteria [33, 34] using the time- qubits are replaced by classical spin vectors M~ i = dependent potential in Eq. (8). Further details are given (sin θi cos φi, sin θi sin φi, cos θi) corresponding to the av- in AppendixB. This evolution can be thought of as in- erage magnetization of a spin-coherent state [30] along cluding the effect of a finite temperature environment in the (x, y, z) directions. The dynamics of the spin vectors the strong coupling limit [35], although here we do not are given by: restrict the evolution to a plane. We show in Fig.4 the dependence on temperature, where we see a strong bias z z d for the M1 = M2 = 1 state at low temperatures. To un- M~ = ωt H~ × M~ , (6) ds i f i i derstand the reason for this bias, we expand the potential 4

z at s = 1 about M1 = 1: 1 1

z z  z 2 V (1) = −1+(1 − M2 ) (1 − M1 )+O (1 − M1 ) , (9) 0.5 0.5 which is minimized only at M z = 1. This is in contrast Populations 2 Populations z to what happens at precisely M1 = 1, where any value z z 0 0 of M2 minimizes the potential at s = 1. Thus, if M1 10 -2 10 0 10 2 10 4 10 -5 10 -3 10 -1 10 1 deviates slightly from 1, which naturally occurs in this Monte Carlo algorithm, there is an energetic bias towards (a) (b) z M2 = 1 [36]. FIG. 5. Computational basis populations for (a) Spin-vector 1 1 dynamics and (b) Spin-vector Monte Carlo with α = 2 and β = 0.05. The results here are equivalent to those of Figs. 3(a) and 4(b). For (b), the simulations use 106 sweeps for each of 0.5 0.5 the 104 independent runs performed. The error bars corre- spond to 2σ confidence intervals generated by 103 bootstraps over the independent runs. 0 0 10 -5 10 -3 10 -1 10 1 10 -5 10 -3 10 -1 10 1

(a) (b) 0.5 0.5 0.4 0.4 FIG. 4. Spin-vector Monte Carlo results for varying temper- 0.3 0.3 atures at α = 2, β = 0.05. (a) Average magnetization along 0.2 0.2

x-axis. (b) Average magnetization along z-axis. The simu- Negativity Negativity lations use 106 sweeps for each of the 104 independent runs 0.1 0.1 performed. The error bars correspond to 2σ confidence inter- 0 0 -2 -1 0 1 2 -2 -1 0 1 2 vals generated by 103 bootstraps over the independent runs. (a) (b) In order to more directly compare to the populations FIG. 6. Negativity at fixed β = 0.05 for (a) κ2 = 0 and of the computational basis states as measured by quan- 4 2 ~ varying tf , and (b) ωtf = 10 and varying κ . Also shown is tum annealing, we use that the spin vector Mi can be the ground state value as calculated from perturbation theory interpreted as a vector on the Bloch sphere, so we can (denoted ‘GS’) . assign the probability of measuring a state |0i for the i- 1 z th qubit to be 2 (1 + Mi ). The results in Figs. 3(a) and 4(b) can then be used to assign probabilities for each of the computational basis states for the 2-qubit system, as from a symmetric state to an antisymmetric state under shown in Fig.5. We see that for the spin-vector dynam- the interchange of the two qubits, and the |00i ground ics, the long time state corresponding to θ~ = (0, π/2) state cannot be part of an antisymmetric combination. at α = 2 corresponds to equal probabilities of finding While our analysis was done in dimensionless units, it the states |00i and |01i. This means that there will al- is useful to give a sense of the parameters values for a rea- ways be a finite non-negligible population on the |00i sonable choice of energy scale for the Hamiltonian. If we state, in contrast to the results of our unitary dynamics. take ω = 1GHz and α = 2, then the closed system evo- Similarly, for Spin-vector Monte Carlo, the strong bias lution shown in Fig. 2(a) becomes effectively adiabatic z z towards M1 = M2 = 1 at low temperatures means we for tf ∼ 10µs with the minimum ground state energy have almost all the population on the |00i state. gap of approximately 0.03GHz at around s = 0.49 (see Fig. 1(a)). For the coupling to a thermal environment at 12mK shown in Fig. 2(a), our results suggest that our V. DISCUSSION proposal requires a high level of coherence if we want to see a complete suppression of the |00i state. However We have proposed a quantum annealing protocol using we see from our simulation results of the spin-vector dy- a two qubit Hamiltonian to validate a tunable antiferro- namics in Figs.3 and4 that the classical analogue fails magnetic XX interaction. The Ising Hamiltonian at the to exhibit this suppression at the same time scales, so end of the anneal has three ground states, and our ex- we may not need an absolute suppression signature to perimental signature is the suppression of one of these clearly distinguish between the quantum dynamics and ground states, the |00i state, for a sufficiently strong at least this semiclassical model. antiferromagnetic XX interaction. Our construction re- It is interesting to note that the strong change in the lies on the instantaneous ground state changing character character of the ground state as a function of α in Fig.1 5 is also associated with a large change in the entanglement quantum annealers with such interactions [26]. of the ground state. We characterize this using the neg- ativity [37], which vanishes for unentangled states and is 1 Γ
ACKNOWLEDGMENTS given by N (ρ) = 2 ||ρ ||1 − 1 , where ρ is the density matrix of the system and ρΓ denotes the partial transpose of ρ with respect to one of the qubits and || · ||1 denotes We thank Daniel Lidar for useful comments on the the trace norm. We show in Fig.6 how the negativity manuscript. Computation for the work described here behaves both as a function of annealing time and system- was supported by the University of Southern California’s bath coupling, where we observe that close to the ideal Center for High-Performance Computing (http://hpcc. case the negativity drops before reaching its maximum usc.edu) and by ARO grant number W911NF1810227. value of as we cross α = 1. The research is based upon work (partially) supported We conclude by emphasizing again that the antisym- by the Office of the Director of National Intelligence metric ground state at large α is impossible with only (ODNI), Intelligence Advanced Research Projects Activ- ferromagnetic XX interactions, since it requires negative ity (IARPA), via the U.S. Army Research Office con- amplitudes in the ground state that can only be gen- tract W911NF-17-C-0050. The U.S. Government is au- erated by non-stoquastic Hamiltonians. While the role thorized to reproduce and distribute reprints for Gov- of such ground states in improving the performance of ernmental purposes notwithstanding any copyright an- quantum annealing remains an open research question, notation thereon. The views and conclusions contained the known examples where such an improvement is pos- herein are those of the authors and should not be inter- sible [19, 20, 23] definitely generate ground states with preted as necessarily representing the official policies or both positive and negative amplitudes [23]. We hope endorsements, either expressed or implied, of the ODNI, our proposal will be relevant for testing next generation IARPA, or the U.S. Government.

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1 1 new spin-vector with angles given by: 0.9 θ = cos−1(2v − 1) , φ = 2πu , (B1) 0.8 0.7 where (u, v) are two uniform random variables on [0,1). 0.6 We calculate the energy difference ∆E according to the 0.5 0.9 potential energy in Eq. (8) in the main text. If ∆E < 0, 0 0.5 1 0 0.5 1 the new spin-vector is accepted; if ∆E > 0, then we draw another uniform random number on (0,1) η, and if (a) (b) η < exp(−β∆E), then the new spin vector is accepted. Otherwise, no update is performed. We discretize our s FIG. 7. Instantaneous ground state population along the an- parameter into nsw identical steps from 0 to 1, and for neal according to the weak-coupling master equation simula- each step we perform a single such update attempt on tions with α = 2, β = 0.05, dimensionless system-bath cou- 2 3 each spin. Therefore, our Monte Carlo algorithm per- pling κ , energy scale kB T/~ω = 1.57 and (a) ωtf = 10 and forms n sweeps, performing a single sweep for each (b) ωt = 5 × 103. sw f increment of the discretized s value, corresponding to a total of 2nsw single-spin update attempts.