Abstract
We formulate and analyze a doubleslit proposal for quantum annealing, which involves observing the probability of finding a twolevel system (TLS) undergoing evolution from a transverse to a longitudinal field in the ground state at the final time t_{f}. We demonstrate that for annealing schedules involving two consecutive diabatic transitions, an interference effect is generated akin to a doubleslit experiment. The observation of oscillations in the ground state probability as a function of t_{f} (before the adiabatic limit sets in) then constitutes a sensitive test of coherence between energy eigenstates. This is further illustrated by analyzing the effect of coupling the TLS to a thermal bath: increasing either the bath temperature or the coupling strength results in a damping of these oscillations. The theoretical tools we introduce significantly simplify the analysis of the generalized LandauZener problem. Furthermore, our analysis connects quantum annealing algorithms exhibiting speedups via the mechanism of coherent diabatic transitions to nearterm experiments with quantum annealing hardware.
Introduction
Feynman famously wrote that the doubleslit interference experiment “… has in it the heart of quantum mechanics. In reality, it contains the only mystery”.^{1} Here we propose a doubleslit experiment for quantum annealing (QA). In analogy to Feynman’s particlewave doubleslit, the proposed experiment can only be explained by the presence of interference and would break down upon either an intermediate measurement or strong decoherence. We are motivated by the recent resurgence of interest in quantum annealing using the transverse field Ising model,^{2,3} which has led to major efforts to build physical quantum annealers for the purpose of solving optimization and sampling problems,^{4,5,6,7} and significant debate as to whether quantum effects are at play in the performance of such devices.^{8,9} The mechanisms by which QA might achieve a speedup over classical computing remain hotly contested, and while tunneling is often promoted as a key ingredient^{10} and entanglement is often viewed as a necessary condition which must be demonstrated,^{11,12} a consensus has yet to emerge. Yet, an explicit example is known where QA theoretically provides an oraclebased exponential quantum speedup over all classical algorithms,^{13} and other examples are known where QA provides a speedup over classical simulated annealing.^{14,15,16,17,18,19} An essential feature in all these cases are diabatic transitions which circumvent adiabatic ground state evolution to enable the speedup, in the spirit of the idea of shortcuts to adiabaticity.^{20,21} When these transitions result in a coherent recombination of the ground state amplitude (a phenomenon known as a diabatic cascade^{16,22}), the result is a wavelike interference pattern in the ground state probability as the anneal time is varied.^{23,24,25} We thus conjecture that coherent recombination of ground state amplitudes after coherent evolution between diabatic transitions can play a critical role in enabling quantum speedups in QA. The doubleslit proposal we formulate and analyze here is designed to test for the presence of quantum interference due to such coherent evolution.
Viewed from a different perspective, our doubleslit proposal joins a family of protocols designed to probe the dynamics of what Berry called the “simplest nonsimple quantum problem”,^{26} a driven TLS near level crossings.^{27} The twolevel paradigm was introduced long ago by Landau and Zener (LZ).^{28,29} The corresponding Hamiltonian for the generalized LZ problem is
where X, Y and Z are the Pauli matrices. In the original protocol which LZ solved analytically, a(t) is constant, b(t) is linear in t, and t runs from −∞ to ∞. The problem has since been studied under numerous variations, including LandauZenerStueckelberg interferometry where b(t) is periodic,^{30,31,32} the subject of various experiments.^{33,34,35,36} Complete analytical solutions were limited until recently to certain particular functional forms of b(t) with constant a(t),^{37} a finiterange linear schedule for both a(t) and b(t),^{38} and periodic a(t) and b(t).^{39} An analytical solution for general b(t) but constant a(t) was found in ref. ^{40}, which was then extended to general (but implicitly specified) a(t) as well.^{41,42} Here we consider the case of general schedules a(t) and b(t), and develop a simple to interpret, yet surprisingly accurate, loworder timedependent perturbation theory approach, that allows us to identify a class of schedules exhibiting “giant” (relative to linear schedules) interference oscillations of the ground state population as a function of the total annealing time. Our proposal should in principle be straightforward to implement using, e.g., flux qubits, and toward this end we also study the effects of coupling to a thermal environment.
The structure of this paper is as follows. In the first section we analyze the TLS quantum annealing problem in the closed system limit. We first transform to an adiabatic interaction picture and perform a Magnus expansion, which allows us to give a simple expression for the ground state probability in terms of the Fourier transform of a key quantity we call the angular progression. We then analyze both the LZ problem (with a linear schedule) and a “Gaussian angular progression” schedule which gives rise to large interference oscillations. We explain how these oscillations can be interpreted in terms of a doubleslit experiment generating interference between ground state amplitudes. In the second section we analyze the problem in the presence of coupling to a thermal environment. We consider the weakcoupling limit both without and with the rotating wave approximation, and find the range of coupling strengths and temperatures over which the interference oscillations are visible, using parameters relevant for superconducting flux qubits. We find a simple semiempirical formula that accurately captures all our opensystem simulation results in terms of three physically intuitive quantities: the oscillation period, rate of convergence to the adiabatic limit, and damping due to coupling to the thermal environment. We express all three are in terms of the input parameters of the theory. Conclusions and the implications of our results are discussed in the final section. A variety of supporting technical calculations and bounds are provided in the Supplementary Information.
Results
We present our results by first considering the closed system setting, followed by the open system case.
Closed system analysis
We first consider the closed system setting. Consider a twolevel system (TLS) quantum annealing Hamiltonian in the standard form (1), where the annealing schedules a(t), b(t) ≥ 0 respectively decrease/increase to/from 0 with time t ∈ [0, t_{f}], where t_{f} is the duration of the anneal. The schedules need not be monotonic, and our analysis thus includes “reverse annealing”^{43,44,45,46,47} as a special case. The TLS can be a single qubit or the two lowest energy levels of a multiqubit or multilevel system separated by a large gap from the rest of the spectrum. Key to our analysis is a series of transformations designed to arrive at a conveniently reparametrized interaction picture. First, we rewrite Eq. (1) in the form
where A(s) = 2a(t)/E_{0} and B(s) = 2b(t)/E_{0} are dimensionless schedules parametrized by the dimensionless time s = t/t_{f}, and E_{0} > 0 is the energy scale of the Hamiltonian. We have cyclically permuted the Pauli matrices for later convenience. The ground states of H_{S}(0) and H_{S}(1) are 0〉 and −i〉, respectively. Second, we parametrize the annealing schedules in the angular form
where θ(0) = 0 and θ(1) = π/2. Under this parametrization the eigenvalues of H_{S}(s) are ±E_{0}Ω(s)/2, so the gap is Δ(s) = E_{0}Ω(s). Thus, any nontrivial timedependence of the gap is encoded in the timedependence of Ω(s), which we refer to as the dimensionless gap. The quantity
is the cumulative dimensionless gap. Third, changing variables from s to τ to absorb Ω(s), the system satisfies the Schrödinger equation
(we work in \(\hbar = 1\) units throughout). The Hamiltonian is diagonalized at each instant by the rotation R_{X}(θ) = e^{−iθX/2}. Thus, fourth, we change into the adiabatic frame^{48,49} with \(\left {\psi _{{\mathrm{ad}}}} \right\rangle = R_X(\theta )\left \psi \right\rangle\), yielding:
We call \(\frac{{d\theta }}{{d\tau }}\) the angular progression of the anneal.
Finally, we transform into the interaction picture with respect to the free Hamiltonian H_{0} = −E_{0}t_{f}Z/2 and its propagator \(U_0(\tau ) = e^{  iH_0\tau }\). Letting S_{±} = (X ± iY)/2 denote the spin raising and lowering operators we have \(X_{\mathrm{I}}(\tau ) = U_0^\dagger (\tau )XU_0(\tau ) = e^{  iE_0t_f\tau }S_ + + {\mathrm{h}}.\,{\mathrm{c}}.\), and obtain
where \(\left {\psi _{\mathrm{I}}} \right\rangle = U_0^\dagger \left {\psi _{{\mathrm{ad}}}} \right\rangle\) and \(\lambda (\tau ) = \frac{1}{2}\frac{{{\mathrm{d}}\theta }}{{{\mathrm{d}}\tau }}\). Therefore, we see that in this adiabatic interaction picture the dynamics of the annealed TLS is a rotation about the timedependent X_{I} axis with a rate equal to the angular progression.
The corresponding timeordered propagator \(U_{\mathrm{I}}(\tau ) = T_{+} e^{i{\int}_{0}^{\tau}\mathrm{d} \tau^{\prime} H_{\mathrm{I}}(\tau^{\prime})}\) can be calculated in timedependent perturbation theory using the Magnus expansion (see Methods) for the Hermitian operator \({\cal{K}}^{(N)}(\tau ) = \sum\nolimits_{n = 1}^N K_n(\tau )\). The resulting \(U_{\mathrm{I}}^{(N)}(\tau ) = {\mathrm{exp}}[  i{\cal{K}}^{(N)}(\tau )]\) converges to U_{I}(τ) uniformly with growing N, and is unitary at all orders.^{50} To first order:
where
To systematically go beyond first order we note that the K_{n}(τ) are nth order nested commutators, and hence closure of the su(2) Lie algebra guarantees that at all orders \({\cal{K}}^{(N)}(\tau ) = \eta ^{(N)}(\tau )\hat n^{(N)}(\tau ) \cdot \vec \sigma\), where η^{(N)}(τ) > 0, \(\hat n^{(N)}(\tau )\) is a unit vector, and \(\vec \sigma = (X,Y,Z)\). It thus follows that
We will be concerned primarily with the probability of remaining in the ground state at the final time, denoted p_{0←0}. Since \(\left {\psi _{\mathrm{I}}}(s) \right\rangle = U_0^\dagger (\tau (s))R_X(\theta (s))\left {\psi (s)} \right\rangle\), we have \(\left {\psi _{\mathrm{I}}(0)} \right\rangle = \left 0 \right\rangle\) and \(\left {\psi _{\mathrm{I}}(1)} \right\rangle \propto  i\left 0 \right\rangle\). Thus, to Nth order:
where the states 0〉 and 1〉 are the initial ground and excited states, and where τ_{f} ≡ τ(1). To first order we find (see Methods for the explicit form of U^{(1)}):
This conceptually elegant result already indicates that quite generally one may expect the ground state probability to oscillate as a function of the anneal time t_{f}, before the adiabatic limit sets in, a conclusion also reached in ref. ^{25} on the basis of either a largegap (nearadiabatic limit) or very small gap (stationary phase approximation) assumption. Our analysis applies for arbitrary gaps.
Having set up the general analysis framework, let us now first consider the simplest annealing schedule, namely a linear interpolation of the type considered in the original LZ problem:^{28,29} A(s) = 1 − s and B(s) = s. To evaluate Eq. (9) we can change the integration variable to s and approximate τ(s) ≈ τ_{f}s in the exponent, yielding \(\phi _{\tau _f}(\omega ) = \frac{1}{2}{\int_0^1} {\mathrm{d}} s\frac{1}{{s^2 + (1  s)^2}}e^{  i\omega \tau _fs}\) for the firstorder Magnus expansion. We compare this to the numerically exact solution in Fig. 1, which shows remarkably good agreement. The simplicity of our Magnus expansion approach should be contrasted with the analytical solution for linear schedules in terms of parabolic cylinder functions.^{38} Also notable is that while a quantum interference pattern is visible, the oscillations are very weak and not controllable (see the insert of Fig. 1). This motivates us to introduce schedules with strong and controllable quantum interference.
Our goal is to identify a family of annealing schedules that generate strong interference between the paths leading to the final ground state, such that “giant” oscillations of the ground state probability can be observed. Therefore we now introduce Gaussian angular progressions.
Suppose that the angular progression is twostep Gaussian, namely, a sum of two Gaussians centered at τ_{f}/2 ± μ (with μ < τ_{f}/2):
Note that \({\int}_0^{\tau _f} \mathrm{d}\tau \frac{{{\mathrm{d}}\theta }}{{{\mathrm{d}}\tau }} = \theta (1)  \theta (0) = \frac{\pi }{2}\), which fixes c. If we assume that \(\alpha \gg 1\) then we may approximate \({\int}_0^{\tau _f}\) by \({\int}_{  \infty }^\infty\) (we bound the approximation error in the Supplementary Information). Thus \(c = \alpha \sqrt \pi /4\) and Eq. (9) yields \(\phi _{\tau _f}(\omega ) = \frac{\pi }{4}e^{  i\omega \tau _f/2}e^{  [\omega /(2\alpha )]^2}{\mathrm{cos}}(\mu \omega )\). Using Eq. (12), to first order the ground state probability is then
The ground state probability thus approaches its adiabatic limit of 1 on a timescale of t_{ad} (set by the Gaussian width), while undergoing damped oscillations with a period of t_{coh}. The oscillations are overdamped when t_{ad} < t_{coh}. In particular, a single Gaussian (μ = 0) can thus not give rise to oscillations.
We plot the ground state probability p_{G}(t_{f}) ≡ p_{0←0} in Fig. 1, for a twostep Gaussian progression with parameters chosen to represent the underdamped case; the associated annealing schedules are shown in Fig. 2 (top). The amplitude of the resulting preadiabatic oscillations seen in Fig. 1 is, as desired, much larger than that associated with the linear schedule. The accuracy of the firstorder Magnus expansion is again striking, especially given its simplicity compared to the analytical solution approaches.^{40,41,42} We give a bound on the firstorder Magnus expansion approximation error in the Supplementary Information.
What is the origin of the oscillations? The answer is an interference effect between the two paths created by the twostep schedule, which enforces a doubleslit or an unbalanced MachZender interferometer scenario, with π/4 beamsplitters: see Fig. 2 (bottom). The first step is a perturbation that generates amplitude in the excited state, while the second step allows for some of this amplitude to recombine with the ground state. The relative phase between the two paths is \(\xi = E_0t_f{\int}_{s_  }^{s_ + } \Omega (s{^{\prime}})\,\mathrm{d}s{^{\prime}}\), which results in oscillations. In Methods we derive this result via a simple interferometertype model that predicts the curve marked DS Γ = 0 in Fig. 1, which is in excellent agreement with the numerically exact result.
A natural question is whether the observation of interference oscillations as a function of t_{f} implies the existence of quantum coherence in the computational basis at t_{f}. We give a formal proof that the answer is affirmative in Methods. An illustration is given in Fig. 1, for the case of dephasing in the instantaneous energy eigenbasis, which is equivalent to performing a measurement in this basis between the two Gaussian steps. The final ground state probability is then the sum of classical conditional probabilities through each beamsplitter, and as expected, the oscillations disappear.
We emphasize that the angular progression
is the sole quantity needed to determine the ground state probability, per Eqs (9) and (12). In particular, per Eq. (15), any transformation of A(s), B(s) and Ω(s) that leaves \(\frac{{{\mathrm{d}}\theta }}{{{\mathrm{d}}\tau }}\) invariant will not affect P_{G} in the closedsystem setting.
Note, furthermore, that specifying the angular progression does not uniquely determine the annealing schedules A(s) and B(s). This is advantageous for practical purposes, since such schedules are typically implemented via arbitrary waveform generators (AWGs) with bandwidth constraints that can be incorporated into the schedule design process. To determine these schedules we need to specify the dimensionless gap Ω(s) and the angular progression \(\frac{{{\mathrm{d}}\theta }}{{{\mathrm{d}}\tau }}\). We can determine τ(s) by solving the differential equation \(\frac{{{\mathrm{d}}\tau }}{{{\mathrm{d}}s}} = \Omega (s)\) subject to the boundary condition τ(0) = 0. Then θ(s) can be determined by solving the differential equation
subject to appropriate boundary conditions. Together, Ω(s) and θ(s) determine the annealing schedules A(s) and B(s) via Eq. (3). In the twostep Gaussian case this means integrating Eq. (13), which, for a constant gap, yields θ(s) as a sum of erf functions.
A particularly interesting example of a dimensionless gap schedule is one that represents the presence of two avoided level crossings, a significant feature of the glued trees problem.^{13} An example is shown in Fig. 2 (top), representing an example of the procedure outlined above for numerical determination of the schedule. It is clear from Eq. (15) that the main contribution to the angular progression is the nearvanishing of the gap. In contrast, when Ω(s) is constant, the main contribution to the angular progression is the suddenness of the schedule, i.e., a large A′(s) or B′(s).
Open system analysis
While a phenomenological model of dephasing in the instantaneous energy eigenbasis already shows clearly how the interference pattern disappears under decoherence (Fig. 1 and Methods), this is not a realistic model of decoherence. We thus examine the effect of coupling the TLS to a thermal environment that corresponds more closely to experiments, e.g., with superconducting flux qubits.
We consider a dephasing model wherein the total systembath Hamiltonian is H = H_{S}(t) + H_{B} + gY ⊗ B, where B is the dimensionless bath operator in the systembath interaction, H_{S}(t) is given in Eq. (2), H_{B} is the bath Hamiltonian, and g is the coupling strength with units of energy. We assume a separable initial state ρ_{S}(0) ⊗ ρ_{B}, with ρ_{B} = exp(−βH_{B})/Z the Gibbs state of the bath at inverse temperature β and partition function Z = Tr[exp(−βH_{B})]. We transform to the interaction picture with respect to H_{B}, so that \(H \mapsto \tilde H(t) = H_S(t) + gY \otimes \tilde B(t)\), with \(\tilde B(t) = U_B^\dagger (t)BU_B(t)\), and \(U_B(t) = e^{  itH_B}\). The same series of transformations as those leading to Eq. (6) can be summarized as: \(Y \otimes \tilde B(t) \,\mapsto\, t_fY \otimes \tilde B(s) \,\mapsto\, t_fR_X(\theta )YR_X(  \theta ) \otimes \tilde B(s) = t_f[{\mathrm{cos}}(\theta )Y + {\mathrm{sin}}(\theta )Z] \otimes \tilde B(s)\). After the final transformation to the H_{0}interaction picture, the total Hamiltonian replacing H_{I}(τ) in Eq. (7) becomes
where \(\vec \mu = ({\mathrm{sin}}\,\phi \,{\mathrm{cos}}\,\theta ,{\mathrm{cos}}\,\phi \,{\mathrm{cos}}\,\theta ,{\mathrm{sin}}\,\theta )\) is a unit vector in polar coordinates, with \(\phi (s) \equiv  E_0t_f\tau (s)\), and henceforth the dot denotes \(\frac{{\mathrm{d}}}{{{\mathrm{d}}s}}\). The timeconvolutionless (TCL) expansion^{51} provides a convenient and systematic way to derive master equations (MEs) without requiring an adiabatic or Markovian approximation. With the detailed derivation given in Methods, the 2nd order TCL (TCL2) ME in the adiabaticframe can be written as:
where
and \(C(s,s^{\prime}) = {\mathrm{Tr}}[\tilde B(s)\tilde B(s^{\prime})\rho _B] = C^ \ast (s^{\prime},s)\) is the bath correlation function. We assume that the bath is Ohmic with spectral density \(J(\omega ) = \eta \omega e^{  \omega /\omega _c}\). To ensure the validity of the TCL2 approximation—which is also known as the Redfield ME—we derive a general error bound in the Supplementary Information, and apply this bound to the Ohmic case. We find the condition \(t_f \ll \frac{\beta }{{g^2\eta }}\), which is always satisfied in our simulations.
In general, the Redfield ME (18) does not generate a completely positive map, which can result in nonsensical results such as negative probabilities.^{52,53} Although this is not necessary for complete positivity,^{54} a further rotating wave approximation (RWA) is usually performed. The resulting Lindbladtype ME also lends itself to a simpler physical interpretation. As detailed in Methods, this leads to
where ρ_{ab} = 〈aρ_{S}b〉, all quantities except g, t_{f} and β are sdependent, and the effective dephasing and thermalization rates γ_{d} and γ_{t}, respectively, and the basis {a〉, b〉}, are given by
Here \(\left {\varepsilon _ \pm (s)} \right\rangle = U_0^\dagger (s)\left \pm \right\rangle\) are the instantaneous eigenvectors of H_{I}(s). The Lamb shift is:
The functions γ(ω)/2 and S(ω) are the real and imaginary parts of the onesided Fourier transform of the bath correlation function, and are implicitly βdependent (see Methods, where we also discuss the validity conditions for the RWA).
The numerical solutions of Eqs (18) and (20) are shown in Fig. 3 for the twostep Gaussian schedule with parameters as in Fig. 1 and for the gap schedule plotted in Fig. 2 (top). The main message conveyed by this figure is that oscillations are visible over a wide range (an order of magnitude) of temperatures and systembath coupling strengths. We also note that for these parameter values the Redfield ME produces physically valid solutions, despite the concerns about complete positivity mentioned above. The Redfield ME results in consistently higher ground state probabilities than the RWA.
These numerical results are accurately reproduced in terms of a simple semiempirical formula, also shown in Fig. 3, and derived in Methods:
where P′_{G}(t_{f}) and P_{G}(t_{f}) denote the open and closed system success probabilities, respectively, where
is the average thermalization rate, and where
is the ground state probability in the adiabatic limit, given by the thermal equilibrium value associated with H_{S}(1) [Eq. (2)]. As seen in Fig. 3, the agreement is excellent with both the RWA result when we use P_{E}(0) = 1/2 (the infinite temperature limit), and with the TCL2 results when we use P_{E}(β) and fit β; we find that the fitted β is consistently slightly lower than the actual β values used in our simulations.
Discussion
We have proposed a doubleslit approach to quantum annealing experiments, exhibiting “giant” interference patterns, motivated by the role of coherent diabatic evolution in enabling quantum speedups. Our analytical approach based on a simple timedependent expansion in the adiabatic interaction picture accurately describes the associated dynamics. The experimental observation of such interference oscillations then becomes a clear and easily testable signature of coherence in the instantaneous energy eigenbasis. The test is simple in principle: it involves a quantum annealing protocol that employs the proposed schedules, with a measurement of only the ground state population as a function of the anneal time t_{f}. When the relative phase between the upper and lower paths to the ground state is randomized, the interference effect is weakened.
To explain these results we proposed an effective model that accurately explains the interference oscillations in terms of a few simple parameters. Namely, upon replacing P_{G}(t_{f}) in Eq. (23) by \(p_{0 \leftarrow 0}^{(1)}(t_f)\) as given in Eq. (14a), the three timescales t_{coh}, t_{ad}, and \(1/\bar \gamma _d\) respectively characterize the oscillation period, Gaussian damping due to approach to the adiabatic limit, and exponential damping due to coupling to the thermal bath. We expressed all three timescales in terms of the input physical parameters of the problem [Eqs (14b) and (24)], and together they completely characterize the oscillations and their damping. It is an interesting problem to try to generalize these results to multilevel systems. We do not expect that the general multilevel system case will be amenable to an analytical treatment of the type we developed here, but under the assumption of a timescale separation which would effectively embed a TLS in a multilevel system due to a large gap to higher excited states, we still expect many of our conclusions and analysis methods to hold. Alternatively, highcontrast interference oscillations have been obtained numerically in multilevel systems with a high degree of symmetry.^{55}
We expect that an experimental test of our “doubleslit” proposal will reveal the predicted interference oscillations for qubits that are sufficiently coherent, such as aluminumbased flux qubits,^{5,6,7} Rydberg atoms,^{56,57} or trapped ions.^{58,59} Such an experiment can be viewed as a necessary condition for quantum annealing implementations of algorithms exhibiting a quantum speedup, e.g., the glued trees problem,^{13} which rely on coherence between energy eigenstates. It appears relevant (if not essential) to use such coherence in order to bypass the common objection that stoquastic quantum annealing or adiabatic quantum computing are subject to, which is that they can be efficiently simulated using the quantum Monte Carlo algorithm when restricted to groundstate evolution (with some known exceptions^{60,61}), due to the absence of a sign problem.^{62,63} Therefore an experimental observation of the quantum interference pattern predicted here will bolster our confidence in the abilities of coherent quantum annealers to one day deliver a quantum speedup.
Methods
Magnus and Dyson series
We repeatedly use the following elementary identity for su(2) angular momentum operators:
Note that the Pauli matrices are related via J_{i} = σ_{i}/2, i ∈ {x, y, z}.
Let us denote the solution of the adiabatic frame Hamiltonian given in Eq. (6) by U_{ad}(τ). The adiabatic interaction picture propagator,
the solution of Eq. (7), can be computed using the Dyson series expansion:
Note that each term in the Dyson series contributes to the ground state amplitude if and only if it is an even power, and likewise to the excitation amplitude if and only if it is an odd power. Consequently, the amplitudes calculated from the Dyson series may not be unitary to a desired precision until the terms are calculated to a high enough order. For this reason we prefer the Magnus expansion,^{50} for which
The first few terms are given by
Using \(U_{\mathrm{I}}^{(N)}(\tau) = \exp\left[i{\cal{K}}^{(N)}(\tau)\right]\) and Eq. (8) we thus find
where we wrote ϕ as a shorthand for ϕ_{τ}(E_{0}t_{f}), and where φ = arg(ϕ). This directly results in Eq. (12).
To compute the second order Magnus term we use \(X_{\mathrm{I}}(\tau ) = e^{  iE_0t_f\tau }S_ + + {\mathrm{h}}.{\mathrm{c}}.\) for the commutation relation
so that
Doubleslit interpretation
Having derived the adiabatic frame Hamiltonian given in Eq. (6)
we see that the angular progression \(\frac{{{\mathrm{d}}\theta }}{{{\mathrm{d}}\tau }}\) of an annealing schedule is the perturbation that causes transitions between the two levels of the system. While this perturbation is steady and small in the case of a linear schedule, Gaussian schedules in which the perturbation is localized suggest an appealing physical picture similar to a doubleslit or interferometer model.
Single Gaussian step
Let us first consider a single Gaussian step, which Eq. (13) reduces to when μ = 0, \(c = \alpha \sqrt \pi /2\). Under the same assumptions as those leading to Eq. (14), we then find \(\phi _{\tau _f}(\omega ) = \frac{\pi }{4}e^{  i\omega \tau _f/2}e^{  (t_f/t_{\mathrm{ad}})^2}\), with ω = E_{0}t_{f}. Thus, Eq. (31) gives us the first order Magnus expansion propagator in the interaction picture with
and ϕ = E_{0}t_{f}τ_{f}/2. The Xrotation matrix in Eq. (31c) thus becomes:
with the superscript G serving as a reminder that this is the Gaussian step case. Now let us suppose that the Gaussian profile is narrow: \(\alpha \gg E_0t_f\), or equivalently \(t_{{\mathrm{ad}}} \gg t_f\). The perturbation is then sudden relative to the adiabatic timescale, and acts like a beamsplitter in a MachZehnder (MZ) interferometer.^{33} In this limit ϕ ≈ π/4 and Eq. (31c) gives
Recall that in the adiabatic interaction picture ψ_{I}(0)〉 = 0〉. Thus, the first phase factor e^{−iϕZ} has no effect, and we can picture a process by which the ground state 0〉 is instantly split into an equal superposition \(\frac{1}{{\sqrt 2 }}\left( {\left 0 \right\rangle  i\left 1 \right\rangle } \right)\) by the “MachZender” matrix \(M_{\pi /2}^{\mathrm{G}}\). These two states are then propagated freely by \(U_0^\dagger (\tau _f) = e^{i(E_0t_f\tau _f/2)Z}\), so they accumulate a relative phase of \(ie^{iE_0t_f\tau _f}\). For a single Gaussian, interference due to this phase difference is clearly not picked up via a Z basis measurement.
Two Gaussian steps: indirect derivation of the interferometer model in the narrow Gaussian limit
If instead we consider a twostep Gaussian schedule [Eq. (13)], then as we already found before Eq. (14), \(\phi _{\tau _f}(\omega ) = \frac{\pi }{4}e^{  i\omega \tau _f/2}e^{  (t_f/t_{\mathrm{ad}})^2}{\mathrm{cos}}(\mu \omega )\), with ω = E_{0}t_{f}. Eq. (31) now gives us the first order Magnus expansion propagator in the interaction picture with \(\phi  = \frac{\pi }{4}{\mathrm{cos}}(\mu E_0t_f)e^{  (t_f/t_{\mathrm{ad}})^2}\) and again ϕ = E_{0}t_{f}τ_{f}/2. Note that without the exponential decay factor \(e^{  (t_f/t_{{\mathrm{ad}}})^2} = e^{  (t_f/t_{{\mathrm{ad}}})^2}\) the oscillations are completely undamped and the adiabatic limit is never reached. Thus it is clear that the finite width of the Gaussian steps is solely responsible for the onset of adiabaticity.
Let us now derive an equivalent MZ interferometer model. On the one hand, we already know from Eq. (12) that \(p_{0 \leftarrow 0}^{(1)} = {\mathrm{cos}}^2(\phi )\), i.e.
This function has a quasiperiod (the distance between consecutive maxima) of π/(μE_{0}), a minimum of cos^{2}(π/4) = 1/2 at t_{f} = 0, and a maximum of 1. On the other hand, we may model the twostep narrow (α ≫ E_{0}t_{f}) Gaussian schedule as two consecutive, localized (at τ_{f}/2 ± μ) and nonoverlapping (α ≫ 1/μ) “beamsplitter” steps, separated by a dimensionless time interval of 2μ. Each beamsplitter is of the form given in Eq. (37), the only difference being that the first acts at τ_{f}/2 − μ (preceded by free evolution) and the second acts at τ_{f}/2 + μ (followed by free evolution). In between the beamsplitter action there is free evolution of duration 2μ. Ignoring the initial and final free evolutions (since the initial and final state we are interested are both 0〉, which is invariant under U_{0}) we expect to be able to write the propagator as the following ansatz:
where we left the angle ψ in the beam splitter matrix (36) unspecified in order to determine it by matching to the properties of \(p_{0 \leftarrow 0}^{(1)} = {\mathrm{cos}}^2(\phi )\). Carrying out the matrix multiplication and computing the expectation value, we find
In order for this to match Eq. (38), we require a quasiperiod of π/(μE_{0}) (which is already the case), a minimum of 1/2 at t_{f} = 0, and a maximum of 1. The latter two conditions force ψ = π/4.
Therefore, considering Eq. (39), we have shown that the twostep Gaussian model is equivalent (in the large α limit) to a MZ interferometer with two unbalanced beamsplitters, separated by free propagation of duration 2μ (the separation between the two Gaussians).
The doubleslit (or MZ interferometer model) is remarkably accurate in terms of predicting the ground state probability. This is shown in Fig. 1, where we compare the numerically exact result and the solution of the simple interferometer model given by Eq. (40). Namely, we use the interference model given in Eq. (40), with ψ = π/4. To calculate the interference fringe, the position of each of the two Gaussians is given by s_{±} = (τ_{f}/2 ± μ)/τ. The phase factor μE_{0}t_{f}, which only holds in the large α limit, is replaced by \(E_0t_f[\tau (s_ + )  \tau (s_  )] = E_0t_f{\int}_{s_  }^{s_ + } {\mathrm{d}} s{^{\prime}}\Omega (s{^{\prime}})\), where τ(s) is the cumulative dimensionless gap [Eq. (4)]. The reason for this replacement is given in the following, alternative and more direct derivation of the interferometer model.
Two Gaussian steps: direct derivation of the interferometer model
Given the twostep Gaussian schedule, Eq. (13),
where τ_{±} = τ_{f}/2 ± μ, we can split the unitary generated by the adiabatic frame Hamiltonian, Eq. (34), into two parts:
We now wish to apply the Magnus expansion separately to each of the unitaries \(U_{{\mathrm{ad}}}\left( {\frac{{\tau _f}}{2},0} \right)\) and \(U_{{\mathrm{ad}}}\left( {\tau _f,\frac{{\tau _f}}{2}} \right)\). Consider \(U_{{\mathrm{ad}}}\left( {\frac{{\tau _f}}{2},0} \right)\). Inverting Eq. (27), the first order Magnus expansion [Eq. (31)] gives
where, using Eq. (9), now
For α ≫ 1 we may extend the limits of integration over the interval [0, τ_{f}/2] to ±∞ without considering the second Gaussian step:
where we used \(c = \alpha \sqrt \pi /4\) as we found in the derivation of Eq. (14). We may thus write the explicit form of the interaction picture unitary as
and the adiabatic frame unitary becomes:
Repeating this calculation for the second adiabatic frame unitary \(U_{{\mathrm{ad}}}\left( {\tau _f,\frac{{\tau _f}}{2}} \right)\), we obtain
Thus, Eq. (42) becomes
which describes an interferometer composed of two unbalanced (π/4) double beamsplitters, interrupted by free propagation of duration τ_{+} − τ_{−} (ignoring the initial and final phases).
The phase accumulated between 0〉 and 1〉 is solely determined by the free evolution in Eq. (49),
whose value is given by
where in the second equality we used Eq. (4).
Interference oscillations in the doubleslit experiment imply quantum coherence in the computational basis
Here we prove that coherence in the energy eigenbasis implies, in general, coherence in the computational basis.
Let H(t) denote an arbitrary, timedependent TLS Hamiltonian, with instantaneous energy eigenbasis {ε_{i}(t)〉}. The TLS density matrix can be written in this basis as
Let us define “coherence” with respect to a given basis as the offdiagonal elements of the density matrix in the same basis. We can compute the coherence in the computational basis {0〉, 1〉} via
where ε_{ij}(t) = ε_{i}(t)〉〈ε_{j}(t). The two bases are related via a unitary rotation:
so that Eq, (53) reduces to:
where we used \(\tilde{\rho}_{00} + \tilde \rho _{11} = 1\) and \(\tilde {\rho} _{01} = \tilde {\rho}_{10}^{\ast}\). Equation (55) can be further simplified using \(\left(\tilde \rho _{00}  \frac{1}{2} \right){\mathrm{sin}}(2\theta )  {\mathrm{Re}}(\tilde \rho _{10}){\mathrm{cos}}(2\theta ) = C({\mathrm{cos}}\,\varphi \,{\mathrm{sin}}(2\theta )  \sin\varphi \,{\mathrm{cos}}(2\theta ))\), where
Additionally, by making use of the trigonometric identity sin(2θ − φ) = sin 2θ cosφ − sinφ cos2θ, Eq. (55) can be written as
Since \(C\sin(2\theta  \varphi )\, \in \,{\Bbb R}\), it follows that \({\mathrm{Im}}(\tilde{\rho}_{10}(t)) \;\ne\; 0\) implies \(\langle 0\rho (t)1\rangle \;\ne\; 0\). Therefore we next establish that indeed, \({\mathrm{Im}}(\tilde \rho _{10}(t)) \;\ne\; 0\) in our doubleslit proposal.
Consider the the ground state just before the first beamsplitter,
with \(\varepsilon /(\tau _ +  \tau _  ) \ll 1\). This state evolves through the doublebeamsplitter region [recall Eq. (49)]:
where U_{0} is given in Eq. (50) and M_{ϕ} is given in Eq. (31d).
After passing through the first beamsplitter, the system density matrix in the energy eigenbasis becomes
It is useful to include a simple model of decoherence between energy eigenstates during the time interval [τ_{−}, τ_{+}], complementary to our master equation treatment. We can do so by introducing a continuous dephasing channel. This damps the phases by the factor e^{−ΓΔτ}, where Δτ = τ_{+} − τ_{−} = 2μ, and Γ > 0 is the dephasing rate. Right before the second beamsplitter, the system density matrix is then:
After passing through the second beamsplitter, the state becomes \(\rho (\tau _ + + \varepsilon ) = M_{\phi }\rho (\tau _ +  \varepsilon )M_{\phi }^\dagger\). We find, after some algebra:
We now note from Eq. (45b) that \(\left \phi \right = \frac{\pi }{8}e^{  (t_f/t_{{\mathrm{ad}}})^2}\). Therefore we may conclude that \({\mathrm{Im}}(\tilde \rho _{10}(t_f)) \,>\, 0\), and \({\mathrm{Im}}(\tilde \rho _{10}) \to 0\) only in the adiabatic limit (\(t_f \gg t_{{\mathrm{ad}}}\), which implies \(\phi  \to 0\)). Note that Eq. (62a) generalizes Eq. (40) by including the effect of dephasing in the energy eigenbasis.
It is clear from Eq. (62) that oscillations in the ground state probability P_{G}(t_{f}), which are present for finite Γ, imply a nonvanishing \({\mathrm{Im}}(\tilde \rho _{10}(t_f))\). Therefore we may conclude that the observation of interference oscillations in our proposed doubleslit experiment are also evidence of coherence in the computational basis at t_{f}. For finite Γ, such coherence vanishes only in the adiabatic limit.
Derivation of the adiabaticframe TCL2/Redfield master equation
We start from the Hamiltonian given in Eq. (17), which we write as
where κ ≡ gt_{f}. Our goal is to derive a master equation for the system evolution. It is convenient to do so using the timeconvolutionless (TCL) approach.^{51} To do so we must first perform yet another interaction picture transformation, defined by H_{I}(s), with the associated unitary \(U_{\mathrm{I}}(s,s{^{\prime}}) = T_ + {\mathrm{exp}}\left[  i{\int}_{s{^{\prime}}}^s H_{\mathrm{I}}(s{^{\prime}}{^{\prime}})ds{^{\prime}}{^{\prime}}\right]\), where T_{+} denotes forward timeordering. In this frame the total Hamiltonian H_{tot}(s) becomes
We can now calculate the TCL expansion generated by the superoperator
whereupon
The different orders are called TCL2, TCL4, etc. We give details on the convergence criteria of this expansion in the Supplementary Information.
To second order the TCL generator is:
where ρ_{B} is the initial state of the bath, and the joint initial state is assumed to be in the factorized form ρ_{S} ⊗ ρ_{B}. Note that the TCL2 approximation coincides with the Redfield master equation.^{51}
Let
denote the bath correlation function. By explicitly tracing out the bath, \({\cal{K}}_2(s)\) can be written as
where
After transforming back to the Schrödinger frame with respect to H_{I}(s) we obtain:
where
Rotating wave approximation
Let
be the onesided Fourier transform of the bath correlation function, where
and where γ_{s}(ω)/2 and S_{s}(ω) are the real and imaginary parts of Γ_{s}(ω). Explicitly:^{51}
Here \({\cal{P}}\) denotes the Cauchy principal value, and the s subscript is a reminder that t_{f} has been factored out.
To perform the rotating wave approximation, let us first define the eigenspace projection operator of H_{I}(s) as
where ε(s)〉 is an eigenstate of H_{I}(s) with instantaneous energy ε(s). We can then define the operator
where
is the dimensionless Bohr frequency, and the sum is over all pairs ε(s), ε′(s) subject to the constraint ε′(s) − ε(s) = ω(s). The interaction picture master equation (66) can then be written to second order, with the TCL2 generator (67) as
To obtain this master equation, we apply the standard Markovian approximation: change the integration variable \(s{^{\prime}} \mapsto s  s{^{\prime}}\) and replace the upper limit with ∞. The RWA consists of neglecting terms in Eq. (79) for which ω′ ≠ ω. A necessary condition for the validity of the RWA is:^{64}
which, unfortunately, is not always satisfied for the twostep Gaussian schedule (13) because [recall Eq. (78)]
for s outside the Gaussian pulse region.
Nevertheless, the RWA results in the interaction picture adiabatic Markovian master equation in Lindblad form:^{65}
where
is the Lamb shift, and
is the dissipator.
We can explicitly calculate A(ω(s)). First, recalling that \(H_{\mathrm{I}}(\tau ) = \frac{1}{2}\frac{{{\mathrm{d}}\theta }}{{{\mathrm{d}}\tau }}U_0^\dagger (\tau )XU_0(\tau )\) [Eq. (7)], we realize that the eigenvalues and eigenvectors of H_{I}(s) can be written as
Also, from the sequence of transformations leading to Eq. (17), the interaction terms have the form
Substituting these expressions back into Eq. (77), we obtain
After undoing the interaction picture transformation with respect to H_{I}(s) and ignoring the phase factors in the A(ω) operators, we obtain the Schrödinger picture master equation, namely Eqs. (20)–(22). In deriving this result we made use of the KuboMartinSchwinger (KMS) condition^{51}
where Δ is the dimensionless Bohr frequency in units of 1/t_{f}:
Derivation of the semiempirical Eq. (23)
The semiempirical formula (23) can be derived directly from Eq. (82). Let us first write Eq. (82) in terms of the quantities defined in Eq. (21b):
We now follow the steps in ref. ^{66} to obtain the solution in this interaction picture. Eq. (90) can be split into two decoupled ordinary differential equations:
where
and
Additionally, the KMS condition allows us to write γ_{d}(s) in terms of \({\cal{F}}_ + (s)\)
The solution of Eqs (91) is given by:
where the initial conditions are:
The next step is to move back to Schrödinger picture
and write the open system ground state probability in terms of \(\tilde \rho _S\):
where
For simplicity, we further denote \(U^a(t) = U_{\mathrm{I}}(t)U_0^\dagger (t)\), whose elements can be related to those of U_{I}(t) in the {0〉, 1〉} basis:
where k, l ∈ {0, 1} and ϕ(t) = −E_{0}t/2. Then:
Because U_{I}(t) is the closed system unitary, we have
and
Eq. (98) becomes:
This result is exact and corresponds to the numerical solution in the TCL2 case shown in Fig. 3.
We now make two additional approximations in order to arrive at a simpler expression. First, we ignore the Lamb shift term Ω(s) in Eqs (95), which leads to:
Second, we substitute the solution given in Eqs (95) into line (104b):
where in the last line we used the weak coupling assumption, \(g^2t_f \ll 1\).
With these two approximations, Eq. (104) becomes the semiempirical formula (23) with P_{E}(0) = 1/2. We note that it is well known that for timeindependent Lindbladians the RWA master equation has the Gibbs state as its steady state.^{51} We do not recover this result for the timedependent case. Rather, we find that the timedependent Redfield master equation (TCL2) converges to the Gibbs state \(P_E(\beta ) = \frac{{e^{\beta E_0/2}}}{Z}\), but with a temperature that differs from that of the bath state, as illustrated in Fig. 3.
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Change history
02 August 2019
An amendment to this paper has been published and can be accessed via a link at the top of the paper.
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Acknowledgements
We are grateful to L. CamposVenuti, L. FryBouriaux, M. Khezri, J. Mozgunov, and P. Warburton for insightful comments and discussions. We used the Julia programming language^{67} and the DifferentialEquations.jl package^{68} for some of the numerical calculations reported in this work. The research is based upon work (partially) supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office contract W911NF17C0050. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.
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D.L. conceived of the project and supervised its execution, and performed initial calculations. H.M.B. performed most of the closed system calculations. H.C. performed most of the open system calculations. H.M.B. and H.C. wrote initial drafts. D.L. wrote the final version with input from H.M.B and H.C.
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MunozBauza, H., Chen, H. & Lidar, D. A doubleslit proposal for quantum annealing. npj Quantum Inf 5, 51 (2019). https://doi.org/10.1038/s4153401901600
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