Robust Diabatic Grover Search by Landau–Zener–Stückelberg Oscillations

Abstract

Quantum computation by the adiabatic theorem requires a slowly-varying Hamiltonian with respect to the spectral gap. We show that the Landau–Zener–Stückelberg oscillation phenomenon, which naturally occurs in quantum two-level systems under non-adiabatic periodic drive, can be exploited to find the ground state of an N-dimensional Grover Hamiltonian. The total runtime of this method is $O(\sqrt{2^n})$, which is equal to the computational time of the Grover algorithm in the quantum circuit model. An additional periodic drive can suppress a large subset of Hamiltonian control errors by using coherent destruction of tunneling, thus outperforming previous algorithms.

1. Introduction

Adiabatic Quantum Computation (AQC) [1,2] is a computational model motivated by the adiabatic theorem. The theorem states that, if a system is prepared in the ground state of an initial Hamiltonian, and the Hamiltonian slowly varies over time, then the system will remain close to the instantaneous ground state [3,4]. By encoding a solution for a computational problem in the ground state of the final Hamiltonian, one can exploit this phenomenon to produce the aforementioned ground state, and thus produce a solution to the problem. The maximal rate of change allowed for such evolution usually scales with the square of the energy gap between the ground state and the first excited state [1].

The Grover problem [5], also known as The Unstructured Search Problem, is one of the few problems solvable by a native adiabatic algorithm, which achieves the same performance as the best possible algorithm in the circuit model [6] (see also [7,8,9]). The input to the problem is an n qubit Hamiltonian $H_p$, which can only be used as a black box, meaning it can be switched on or off:

$$H_p=I_N-|y\rangle \langle y|,$$

(1)

wherein $I_N$ is the $N\times N$ identity matrix with $N=2^n$, and we use dimensionless units (see Appendix A). The problem is to find the unknown string y. Grover’s algorithm solves the problem in time $O(\sqrt{N})$, which is a quadratic improvement to any classical algorithm. The problem is comparable to finding the ground state of a known Hamiltonian, whose ground state is computationally hard to find and, therefore, can be considered computationally unknown [10].

An adiabatic algorithm for the search problem was first suggested by Farhi et al. [1]. The system is initialized to a symmetric superposition of the computational basis states, denoted $|u\rangle =|+\cdots +\rangle$, and then evolves by the time-dependent Hamiltonian:

$$H_G\left(s(t)\right)=(1-s(t))·(I_N-|u\rangle \langle u|)+s(t)·(I_N-|y\rangle \langle y|),$$

(2)

wherein the control function $s(t):[t_i,t_f]\to [0,1]$ is initialized to 0 and increases monotonically in time to 1. The minimal gap for n qubit systems is $\Delta =\sqrt{2^{-n}}$. Evolving with a linear $s(t)$ requires $O(2^n)$ time, while a specially tailored control function, whose rate matches the instantaneous spectral gap, generates the ground state of $H_p$ in the optimal time, $O(\sqrt{2^n})$ [11,12].

In this work, we introduce a diabatic algorithm for the Grover problem, denoted algorithm $\mathcal{A}$, whose performance matches the optimal adiabatic and circuit model algorithms [5,6,12], by setting $s(t)=(1-Acos(\omega t))/2$ when $\omega \gg \Delta$. The system passes the minimal gap multiple times diabatically and is effectively evolving by a Landau–Zener–Stükelberg (LZS) Hamiltonian [13,14,15,16,17] (which is sometimes referred to as the Landau–Zener–Stükelberg–Majorana Hamiltonian [18]). In algorithm $\mathcal{B}$, we add an oscillating term $Bcos(\omega t)\left|u\right.〉\left.〈u\right|$ that yields improved robustness to Hamiltonian control errors relative to previous algorithms [5,12].

2. Background

2.1. LZS Hamiltonians

We start by analyzing the LZS Hamiltonian for a generic two-level system with the bare states $|0\rangle ,|1\rangle$, closely following the treatment of [19]:

$$\begin{array}{c}\hfill H_{\mathrm{LZS}}(t):=\frac{1}{2}\left(-Acos(\omega t){\sigma }_z-\Delta {\sigma }_x\right)=\frac{1}{2}\left[\begin{array}{cc}-Acos(\omega t)& -\Delta \\ -\Delta & Acos(\omega t)\end{array}\right].\end{array}$$

(3)

The sinusoidal drive causes the Hamiltonian to exhibit avoided level crossings at $t=\pi (k+\frac{1}{2})/\omega$ for $k\in \mathbb{N}$ with a minimal energy gap of $\Delta$ (see Figure 1).

To gain some intuition, consider a system initialized to the state $|0\rangle$ and driven through the avoided crossing twice (meaning, one period of $s(t)$). After the double crossing, the population of the state $|1\rangle$, denoted $P_{+}^{(2)}$, approaches 0 for both $\omega \ll {\Delta }^2/A$ and for $\omega \gg A$: if $\omega \ll {\Delta }^2/A$, the adiabatic condition holds; the system always remains in the instantaneous ground state and, thus, returns to $|0\rangle$. Furthermore, in the limit $\omega \gg A$, the propagator approaches unity and the state remains unperturbed. In intermediate cases, an interesting phenomenon occurs: in the first passage of the avoided crossing, the system transfers almost perfectly from the initial ground state to the final excited state. However, a tiny amplitude leaks to the orthogonal state. The populations of the excited state and the ground state gain different phases between the two crossings, and, finally, interfere again in the second crossing. $P_{+}^{(2)}$ is affected by this interference and oscillates with the periodicity of the control $2\pi /\omega$ in what is known as Landau–Zener–Stückelberg oscillations [13,14,15] (See Figure 2).

In the $\omega \gg \Delta$ regime, one can use the rotating wave approximation (see [19], and Appendix B) to show that, with periodic drive, the system oscillates around the x axis in the Bloch sphere with frequency

$${\Omega }=\Delta \left|J_0\left(\frac{A}{\omega }\right)\right|.$$

(4)

The algorithm will fail when $A/\omega$ equals a root of the Bessel function $J_0$, where a coherent destruction of tunneling (CDT) occurs, and ${\Omega }=0$ ([20], see also [17,19]). CDT was previously suggested as a method to control interactions in quantum systems [21,22,23] and we use these ideas in algorithm $\mathcal{B}$.

2.2. Grover as a Two-Level System

Interestingly, the Grover Hamiltonian $H_{\mathrm{G}}(t)$ with a periodic control function is closely related to $H_{\mathrm{LZS}}(t)$. The key to the mapping is the subspace $V=\mathrm{span}\left\{|u\rangle ,|y\rangle \right\}$, which is invariant to $H_{\mathrm{G}}(s)$ for all s, as originally noted by Farhi and Gutmann [24] (see proof in Appendix C). Although V is isomorphic to the Hilbert space of a two-level system, one cannot map $|u\rangle ,|y\rangle$ to $|0\rangle ,|1\rangle$ trivially in $H_{\mathrm{LZS}}$, because the first pair is only approximately orthogonal. To overcome this problem, we define a new basis $|\overline{0}\rangle ,|\overline{1}\rangle$, exponentially close to $|u\rangle$ and $|y\rangle$, as stated in the following claim:

Claim 1.

The projection of $H_{\mathrm{G}}(s(t))$ on V satisfies:

$$H_{\mathrm{G}}(s(t)){|}_V=\frac{I_2}{2}+\left(s(t)-\frac{1}{2}\right)\sqrt{1-{\Delta }^2}{\overline{\sigma }}_z-\frac{\Delta }{2}{\overline{\sigma }}_x,$$

(5)

wherein $\Delta =\langle y|u\rangle$. The operators ${\overline{\sigma }}_x,{\overline{\sigma }}_z$ act on the states

$$\begin{array}{cc}\hfill |\overline{0}\rangle & =\sqrt{\frac{1+\sqrt{1-{\Delta }^2}}{2}}|u\rangle -\sqrt{\frac{1-\sqrt{1-{\Delta }^2}}{2}}|u^{\perp }\rangle \hfill \\ \hfill |\overline{1}\rangle & =\sqrt{\frac{1-\sqrt{1-{\Delta }^2}}{2}}|u\rangle +\sqrt{\frac{1+\sqrt{1-{\Delta }^2}}{2}}|u^{\perp }\rangle ,\hfill \end{array}$$

(6)

wherein $|u^{\perp }\rangle :=\frac{|y\rangle -\Delta |u\rangle }{\sqrt{1-{\Delta }^2}}$ is the vector orthogonal to $|u\rangle$ in V.

See Appendix D for the proof.

3. Results

3.1. Algorithm $\mathcal{A}$

Algorithm $\mathcal{A}$ is an immediate corollary of Claim 1. The Hamiltonian $H_{\mathrm{G}}$ with the control function $s(t)=(1-Acos(\omega t))/2$ acts on V as an LZS Hamiltonian on the states $|\overline{0}\rangle ,|\overline{1}\rangle$. Since $|\overline{0}\rangle$ and $|\overline{1}\rangle$ are exponentially close to $|u\rangle$ and $|y\rangle$, respectively, evolving $|u\rangle$ by $H_{\mathrm{G}}(s(t))$ will cause the system to oscillate between the states close to $|u\rangle$ and $|y\rangle$ with frequency ${\Omega }=\Delta \left|J_0(\sqrt{1-{\Delta }^2}A/\omega )\right|$. Hence, such a driven Hamiltonian can solve the Grover problem in time $O(\sqrt{2^n})$, i.e., the same complexity as the optimal circuit and adiabatic models.

A careful analysis of LZS interferometry shows that the algorithm finds y for a wide range of $A,\omega$. We require only $\omega \gg \Delta$ for the rotating wave approximation to hold. $J_0(\sqrt{1-{\Delta }^2}A/\omega )$ is a factor of the algorithm’s runtime, hence $A/\omega$ should not be large (for $z\gg 1$, $|J_0(z)|\sim 1/\sqrt{z}$), and not too close to the roots of $J_0$ as it will cause ${\Omega }$ to diminish by CDT. Note that none of these constraints requires prior knowledge of the gap $\Delta$, other than an upper bound. Hence, the algorithm is robust to a multiplicative error of the Hamiltonian due to calibration errors, similar to previous approaches [25]. A different choice of parameters may improve the algorithm’s robustness to the variations in the total evolution time, as demonstrated in [26].

The limit $A=0$ yields maximal ${\Omega }$, and corresponds to evolving by the time-independent Hamiltonian $H_G(s){|}_{s=1/2}=\frac{1}{2}(I_2-\Delta {\overline{\sigma }}_x)$, which we denote $H_{1/2}$. This is exactly the “analog” algorithm for the search problem by Farhi and Gutmann [24], and, more generally, a search by a quantum walk [25,27,28,29,30,31,32,33]. The Hamiltonian $H_{1/2}$ is the core of algorithms for the search problem: in the adiabatic algorithm [12], the Hamiltonian spends most of the time close to $H_{1/2}$, where the gap is minimal [25], while the original gate-model algorithm by Grover [5] can be seen as a simulation (or an approximation by Trotter formula [34]) of $H_{1/2}$ [24].

3.2. Algorithm $\mathcal{B}$

We now discuss adding an additional modulation to Algorithm $\mathcal{A}$ to improve its robustness without significantly increasing the runtime. Equation (7) defines the Hamiltonian for algorithm $\mathcal{B}$, and the spectrum is illustrated in Figure 3.

$$\begin{array}{cc}\hfill H_{\mathcal{B}}(t)& =(I_N-|u\rangle \langle u|)·\frac{1+Acos(\omega t)}{2}+(I_N-|y\rangle \langle y|)·\frac{1-Acos(\omega t)}{2}-Bcos(\omega t)\left|u\right.〉\left.〈u\right|\hfill \\ \hfill H_{\mathcal{B}}{|}_V& =\left[\begin{array}{cc}\frac{1}{2}-(B+\frac{A}{2})cos(\omega t)& -\frac{\Delta }{2}(Bcos(\omega t)+1)\\ -\frac{\Delta }{2}(Bcos(\omega t)+1)& \frac{1}{2}+\frac{A}{2}cos(\omega t)\end{array}\right]+O({\Delta }^2).\hfill \end{array}$$

(7)

A natural question is whether Algorithm $\mathcal{B}$ is “cheating” by artificially increasing the gap or by manipulating resources. Here, a small detour discussing resources is in order. First, note that implementing the term $\left|u\right.〉\left.〈u\right|$ requires no prior knowledge of y. Namely the algorithm is the same for all y (or y is “unknown”). This means that the total time wherein $H_p$ is active would have to be at least $2^{n/2}$. Otherwise, this would contradict the optimality of Grover’s algorithm [35]. To understand the role of B, one can partition $H_{\mathcal{B}}$ by the Trotter approximation to slices of time independent Hamiltonians, where evolution by $H_p$ and by terms that are not $H_p$ alternate. In this picture, increasing $\left|B\right|$ corresponds to using a stronger quantum computer between calls to the black box, but has no effect on the query complexity of the problem (the total time $H_p$ is active).

3.3. Robustness Comparison

In what follows, we compare the robustness to control errors of algorithm $\mathcal{B}$ versus applying the time-independent Hamiltonian $H_{1/2}$, which also represents the standard gate model and adiabatic algorithms (for an analysis of the Hamiltonian-based search algorithm under different noise models, cf. [25,36,37]).

Hamiltonian control errors are uncontrolled terms causing the system to deviate unitarily from the intended evolution. The first error we focus on is in the form $A_{1}cos({\omega }_{1}t+\phi ){\overline{\sigma }}_z$ that preserves the subspace V and represents an error in $s(t)$ (see Equation (5)).

Consider $H_{1/2}$ with a harmonic control error in $s(t)$:

$${\tilde{H}}_{1/2}=\frac{I_2}{2}-\frac{\Delta }{2}{\overline{\sigma }}_x+A_{1}cos({\omega }_{1}t+\phi ){\overline{\sigma }}_z.$$

(8)

This is exactly the LZS Hamiltonian. Therefore, for high frequency errors (${\omega }_1\gg \Delta$), the Rabi frequency is $\tilde{{\Omega }}=\Delta \left|J_0\left(\frac{A_1}{{\omega }_1}\right)\right|$, and the evolution is generally unaffected. On the other hand, for ${\omega }_1=0$, even $A_1\approx \Delta$ may cause the system to freeze in the initial state because the ${\overline{\sigma }}_z$ rotation may become more dominant than the desired ${\overline{\sigma }}_x$ rotation. Hence, algorithms based on $H_{1/2}$ are not robust to low frequency control errors.

Algorithm $\mathcal{B}$ generally shows similar robustness (see Figure 4). It fails to find y when ${\omega }_1=0$ and $A_1\approx \Delta$ for the same reasons $H_{1/2}$ fails. For high frequency errors, we write the Hamiltonian $H_{\mathcal{B}}+A_{1}cos({\omega }_{1}t+\phi ){\overline{\sigma }}_z$ in the appropriate rotating frame (around ${\overline{\sigma }}_z$) while neglecting $O({\Delta }^2)$ terms:

$$\begin{array}{cc}\hfill & {\tilde{H}}_{\mathcal{B}}^{\prime }{|}_V=\left[\begin{array}{cc}0& -\frac{\Delta }{2}(Bcos(\omega t)+1)\chi \\ \\ -\frac{\Delta }{2}(Bcos(\omega t)+1){\chi }^{*}& 0\end{array}\right]\hfill \\ \hfill & \chi =\sum _{k,k_1=-\infty }^{\infty }{J}_k\left(\frac{A+B}{\omega }\right)J_{k_1}\left(\frac{2A_1}{{\omega }_1}\right)e^{i{k}_1({\omega }_{1}t+\phi )-ik\omega t}.\hfill \end{array}$$

See Appendix E for details.

The algorithm is generally unaffected by high frequency errors (${\omega }_1\gg \Delta$) where all terms except $k=k_1=0$ average out, and the Rabi oscillation is $\tilde{{\Omega }}=J_0\left(\frac{A+B}{\omega }\right)J_0\left(\frac{2A_1}{{\omega }_1}\right)$. Note that, if $k_1{\omega }_1\approx k\omega$ for some values of $k,k_1$, then these terms would not average out, and may, in principle, cause the algorithm to fail because of CDT.

The second error we consider in our comparison are errors that do not preserve V. For their analysis, we use a three-level system toy model, composed of the previously defined states $|\overline{0}\rangle ,|\overline{1}\rangle$ and an additional state $|\overline{2}\rangle$, which represents a state outside of V. The error term we choose to focus on is the term $\eta (\left|\overline{0}\right.〉\left.〈\overline{2}\right|+\left|\overline{2}\right.〉\left.〈\overline{0}\right|)$. The Hamiltonians take the form:

$$\begin{array}{c}\hfill H_{1/2}=\left[\begin{array}{ccc}\frac{1}{2}& -\frac{\Delta }{2}& \eta \\ -\frac{\Delta }{2}& \frac{1}{2}& 0\\ \eta & 0& 1\end{array}\right]H_{\mathcal{B}}=\left[\begin{array}{ccc}\frac{1}{2}-(B+\frac{A}{2})cos(\omega t)& -\frac{\Delta }{2}(Bcos(\omega t)+1)& \eta \\ -\frac{\Delta }{2}(Bcos(\omega t)+1)& \frac{1}{2}+\frac{A}{2}cos(\omega t)& 0\\ \eta & 0& 1\end{array}\right].\end{array}$$

(9)

Interestingly, $H_{1/2}$ already has some inherent robustness to errors diverting the system to $|\overline{2}\rangle$: the diagonal elements of $H_{1/2}$ can be considered as “potential energies” of three sites. Therefore, a particle in $|\overline{0}\rangle$ needs to overcome a potential difference to reach $|\overline{2}\rangle$, while it does not need to face a barrier when transitioning to $|\overline{1}\rangle$.

Algorithm $\mathcal{B}$ improves the natural error suppression by adding CDT between the states $|\overline{0}\rangle$ and $|\overline{2}\rangle$, while allowing transitions between $|\overline{0}\rangle$ and $|\overline{1}\rangle$. We hereby present a simplified analysis, using the rotating wave approximation. However, we stress that finer tools, such as Floquet theory [38], better describe the dynamics of the system and should be used when one attempts to find optimal values for $B,A,\omega$ (see Figure 5). After changing to a rotating frame where $\langle \overline{0}|H_{\mathcal{B}}|\overline{0}\rangle =\langle \overline{1}|H_{\mathcal{B}}|\overline{1}\rangle =0$, and using the rotating wave approximation, we have:

$$H_{\mathcal{B}}^{\prime }=\left[\begin{array}{ccc}0& -\frac{\Delta }{2}{J}_0\left(\frac{B+A}{\omega }\right)& \eta J_0\left(\frac{B+A/2}{\omega }\right)\\ -\frac{\Delta }{2}{J}_0\left(\frac{B+A}{\omega }\right)& 0& 0\\ \eta J_0\left(\frac{B+A/2}{\omega }\right)& 0& \frac{1}{2}\end{array}\right].$$

(10)

Note that, in Equation (10), the argument for the Bessel function in $\langle \overline{0}|H_{\mathcal{B}}|\overline{1}\rangle$ is different from the one in $\langle \overline{0}|H_{\mathcal{B}}|\overline{2}\rangle$. By choosing $\frac{B+A/2}{\omega }$ to be a root of $J_0$, the transition from $|\overline{0}\rangle$ to $|\overline{2}\rangle$ is suppressed. On the other hand, the term $\left|\langle \overline{0}|H_{\mathcal{B}}|\overline{1}\rangle \right|$, which dominates ${\Omega }$ and the computation time, is multiplied by a factor of $J_0\left(\frac{B+A}{\omega }\right)\ne 0$. Namely, the runtime is increased by a factor of $1/J_0\left(\frac{B+A}{\omega }\right)$, while the error term $\eta$ is suppressed. Our numerical simulations (see Figure 5) confirm that, for a given scenario, the probability of an evolution $H_{1/2}$ to find y is $3.2\times {10}^{-5}$, while, with tuned parameters, Algorithm $\mathcal{B}$ finds y with near certainty.

Thermal noise: Implementing error correction for quantum algorithms based on continuous Hamiltonians instead of discrete gates is an open problem [39]. One can suppress thermal noise (as well as control errors) by encoding the Hamiltonian by a stabilizer code [40], combined with dynamical decoupling [41], energy gap protection [42], or Zeno effect suppression [43]. All of them function very similarly [39,44], providing enhanced performance for finite size systems, which were recently described as noisy intermediate scale quantum (NISQ) [45]. For an exponential time algorithm, such as the unstructured search problem, error suppression methods ultimately fail, and an active error correction is required.

4. Discussion

In this work, we propose a new diabatic algorithm for solving the Grover problem using LZS interferometry. Diabatic scheduling for computation was researched recently [46,47] and, in some cases, even showed superior performances to adiabatic scheduling. Variational Quantum Algorithms (VQA) [48,49,50,51] and Quantum Approximate Optimization Algorithms (QAOA) [52,53], in which the path is optimized by a classical computer, can also be considered diabatic. However, our work shows a new application of diabaticity: suppression of control errors. We conjecture the need for hybrid algorithms (diabatic/adiabatic), tailored to the noise parameters of a system.

One possible near term implementation of such model is the unstructured search problem on the hypercube [30,54], which exhibits an avoided crossing between the two lowest eigenvalues, while the rest of the spectrum is separated. Driving the system with oscillations that are slow with respect to the gap between the first and second excited states will cause the two lowest energy states to act as a two-level system.

While the Grover problem is important on its own, it is interesting to examine the applicability of our paradigm to additional problems. It remains an open question whether one can translate any adiabatic algorithm to a diabatic algorithm.

Finally, it is interesting to find an expression for the optimal driving frequencies in Algorithm $\mathcal{B}$, including their spectral width, and effectiveness (based on a full numerical analysis of the Floquet problem).