Steering random spin systems to speed up the quantum adiabatic algorithm

There is a furious race underway to construct the first practical quantum computer. To complement this, there is a large research effort to broaden the class of problems that can be attacked by these machines. A very promising direction is optimization problems. One of the leading candidate methods for solving such problems on a quantum computer is the quantum adiabatic algorithm (QAA) Farhi et al. (2001), in which the ground state of a simple quantum system is slowly transformed into the solution of the optimization problem. There have been extensive studies of the QAA on classical computers McGeoch (2014) and open-system quantum annealing devices intended to solve similar problems have been constructed Johnson et al. (2011); Boixo et al. (2013, 2014). The QAA exploits the adiabatic theorem and uses the fact that the ground state of appropriate quantum Hamiltonians correspond to difficult classical optimization problems, for which the standard classical search algorithms are inefficient due to the complicated landscape for the cost function Tanaka et al. (2017); Lucas (2014). The difficulty in demonstrating the QAA is the presence of small energy gaps that can lead to generalized Landau-Zener-Stueckelberg-Majorana (LZSM) tunneling Landau (1932); Zener (1932); Stueckelberg (1933); Majorana (1932). Once the tunneling occurs, the system leaves the instantaneous ground state, probably for good, and the algorithm breaks down.

In spin models, we may look more closely at the degrees of freedom that produce the dangerous avoided crossings. The classic LZSM problem can be thought of as a single spin-1/2 particle in a time-dependent magnetic field that reverses the spin direction. This is the local single-particle case. In the other limit, we may imagine a crossing of two levels whose energies are very close, but whose spatial configurations differ by the rearrangement of many spins, perhaps well-separated in space. This is the non-local case. Both contribute to unwanted tunneling.

In this paper, we propose a modification of the QAA that largely eliminates local LZSM tunneling. This modification requires accurate control of individual qubits that was demonstrated recently in various systems, including trapped ions Vittorini et al. (2014), Rydberg atoms Maller et al. (2015) and superconducting qubits Chow et al. (2009). In the conventional annealing protocol, the system is prepared in a strong field along the $x-$direction without interaction, then the field is slowly changed to the final field and the interaction is turned on. During this process, a time-dependent gauge term causes transitions between the instantaneous eigenstates of the Hamiltonian. This term is proportional to the Berry curvature Berry (1984); Avron et al. (2011a, b); Gritsev and Polkovnikov (2012); Xu et al. (2014) and its effect was recently investigated in superconducting devices with a single qubit Schroer et al. (2014) and interacting qubits Roushan et al. (2014). We demonstrate that with the proper compensation of this topological term, qubits acquire protection against excitation processes, increasing the probability for the system to remain in the ground state even for short annealing times. This approach may also point the way toward more general improvements of quantum adiabatic algorithms.

The Hamiltonian in our approach is defined on the time interval $0\leq t\leq t_{a}$, where $t_{a}$ is the annealing time and it has the form:

Here $H_{i}$ and $H_{f}$ are time-independent Hamiltonians that represent a simple problem and a difficult optimization problem, respectively. The scalar functions $f_{i}$ and $f_{f}$ satisfy the boundary conditions: $f_{i}(0)=f_{f}(1)=1$ and $f_{i}(1)=f_{f}(0)=0$. However, we adjust these functions rather than choosing the customary linear-in-time forms. $H_{s}$ is the steering term and key to our approach. The idea of adding an additional term to the Hamiltonian is not new and has been used to convert a stoquastic Hamiltonian to a non-stoquastic Hamiltonian Hormozi et al. (2017); Vinci and Lidar (2017), while modifications to the annealing schedule have been used to add quantum fluctuations Kadowaki and Nishimori (1998). It has also been used in the method of shortcuts to adiabaticity and quantum critical points del Campo et al. (2012); Takahashi (2013); del Campo (2013); Saberi et al. (2014); Del Campo and Zurek (2014); del Campo and Sengupta (2015); Rams et al. (2016); Takahashi (2017); Sels and Polkovnikov (2017). Our method is to make a local approximation to the exact formula for the counterdiabatic driving Hamiltonian, defined in the following paragraph.

We construct $H_{s}$ using a result from adiabatic population transfer theory and counterdiabatic driving Demirplak and Rice (2003); Berry (2009). If a time-dependent Hamiltonian $H_{0}$ has instantaneous eigenstates $|n(t)\rangle$ such that $H_{0}(t)|n(t)\rangle=E_{n}(t)|n(t)\rangle$, then we can define the steering Hamiltonian as

The modified Hamiltonian $H(t)=H_{0}(t)+H_{1}(t)$ drives the ground state $|1\rangle$ of $H_{0}$ without any transitions. If the initial state at $t=0$ is the ground state of $H_{0}$, then the solution of the time-dependent Schrödinger equation at $t_{a}$ is the ground state of $H_{0}$. We could take $H_{0}=f_{i}(t/t_{a})H_{i}+f_{f}(t/t_{a})H_{f}$ and $H_{s}=H_{1}$, and this would yield the solution of the optimization problem with certainty, but unfortunately the computation of $H_{1}$ is not efficient. Instead, we propose a local approximation to $H_{1}$. We note that for single spin-1/2 particle at site $k$ with Hamiltonian $H_{0}^{(k)}(t)=\bm{B^{(k)}}(t)\cdot{\bm{\sigma^{(k)}}}/2$ the steering term is

and we may correct for an arbitrary random magnetic field on an array of spins by summing over $k$.

To illustrate our method we choose the one-dimensional random-field Ising model (RFIM) on a ring of $L$ spins:

with periodic boundary conditions understood. The $h_{k}$ are chosen uniformly from the interval $[-1,1]$. The width of the disorder distribution sets the energy scale. The initial Hamiltonian is chosen as usual to be a uniform transverse magnetic field

In the calculations below we take $h_{0}=10$.

The RFIM at $J=0$ has the simple solution $\langle{\sigma_{z}^{k}}\rangle=-h_{k}/|h_{k}|$, while the $J\rightarrow\infty$ limit is an antiferromagnet. At small $J$, $J<<h_{av}$ ($h_{av}$, average random field, $\sim$ 1/2 in this paper), the ground state has just a few spins that deviate from the $J=0$ solution at sites $k$ where $|h_{k}|$ happens to be small. The spin at site $k$ feels a time-dependent effective field with a $z$-component given by the sum of $h_{k}$ and $J[\langle{\sigma_{z}^{(k-1)}(t)}\rangle+\langle{\sigma_{z}^{(k+1)}(t)}\rangle]$, where $\langle{\sigma_{z}^{(k\pm 1)}(t)}\rangle$ are the time-dependent expectation values of the $z$-components of the neighboring spins. When the magnitude of the total effective field (including the $x$-component) becomes small, the gap becomes small and the QAA can fail. This is the type of failure that our local approximation for $H_{1}$ should be able to fix. At larger $J$ values, ($J$ of order 1) there will be larger clusters of spins that deviate from the $J=0$ solution. This will create situations where there are small energy gaps separating states that differ by many spin flips. Our single-spin approximation for the steering term is then not expected to work, and more sophisticated approximations are required. We will later present a cluster method that is a step in this direction.

It is clear that the steering method is applicable in principle to any model that includes a random field. Our choice of the RFIM is motivated by the facts that it has a relatively small number of parameters, is simple to simulate numerically, and the statistical properties of the final Hamiltonian of Eq. (4) have been well studied. By the standards of the field, the one-dimensional RFIM is fairly simple but it has nevertheless served as a common testbed for the QAA.

Notice that $H_{i}$ and $H_{f}$ are both stoquastic Bravyi and Hastings (2017) but the introduction of $H_{s}$ makes the Hamiltonian non-stoquastic. This is somewhat similar to a previous study, Hormozi et al. (2017), but our motivation for introducing the additional term is quite different.

We choose $f_{i}(t)=\cos^{2}(\pi\tau/2)$ and $f_{f}(t)=\sin^{2}(\pi\tau/2)$, where $\tau\equiv t/t_{a}$. The initial behavior of $f_{f}$ and the final behavior of $f_{i}$ are quadratic; this is chosen so that $H_{s}(t=0)=H_{s}(t=t_{a})=0$ and the derivatives provide slow start and stop. These choices, together with Eq. (3), give

Since $t_{a}$ can be small, the size of the steering term can be large. Of course an arbitrarily large $H_{s}$ is unphysical. Ultimately, the interesting parameter range for the QAA is when $t_{a}$ is large. In this case the steering term is typically small compared to the other terms in the Hamiltonian.

With these definitions we solve the time dependent Schrödinger equation for $H_{qaa}$ numerically Johansson et al. (2012, 2013). For comparison purposes it is useful to solve the same instance of the problem with the above definition of $H_{s}$ (“with steering”) and setting $H_{s}=0$ (“without steering”). We also define the success probability, i.e., the probability to be in the ground state at the end of the evolution, as $P_{1}=|\langle{1}|{\psi(t=t_{a})}\rangle|^{2}$.

We have demonstrated significant improvements in the QAA for random-field spin systems with relatively weak interactions. This is done by adding a term to the Hamiltonian that suppresses transitions representing local spin re-orientations. When the interactions become stronger, the low-energy eigenstates become more extended and the technique in the approximation used here becomes ineffective. In other words, the method is good for insulating phases and not for metallic phases of disordered systems. However, the steering concept itself, as represented by the correction term in Eq. (3) is not at all limited to local modifications of the problem. We have made a cluster expansion to construct a less local form of the operator in Eq. (6). It should also be possible to work out ways of improving the steering so that it is effective in metallic phases as well.

We have not yet investigated systematically the crucial question of how the improvements in the algorithm scale with system size. The local nature of the improvements of the steering would suggest at least a constant speedup comparing to the standard annealing procedure. Of course in practical calculations even a constant speedup is very desirable, as long as the constant is big. For certain problems, we have shown that two orders of magnitude can be achieved.

The protocol is applied to a particular configuration of the final Hamiltonian, where both local fields and couplings between the spins are exactly determined by the corresponding classical optimization problem. To evaluate the performance of the algorithm for different problems with similar structure, we assume that the optimization problems represent an ensemble of random Hamiltonians. The success is determined both by the quantum fidelity of the final state and by the fraction of successful solutions out of the ensemble. For $t_{a}$ fixed at 1, we have the following comparisons for the standard and steered QAAs. Out of exponentially large system size $2^{L}$ with $L=12$, we find with probability above 99% that the system is in one of 21 low energy states when $J=0.1$. For $J=0.3$ and the same $L$, we find one of the 398 low energy states with probability above 99% for the QAA with 1-spin steering. For the unsteered algorithm, the corresponding values are too large — 3949 and 3929, respectively. For the QAA with 1-spin steering, the probability to find one of the lowest 1% of $2^{L}$ (with $L=12$) energy states is 99.7% when $J=0.1$. When $J=0.3$, the probability becomes 81%. For the unsteered algorithm, the corresponding probabilities are only 3% and 4%, respectively. Thus, by controlling $3L$ local fields, we are guaranteed to find one of the low energy states out of $2^{L}$ states.

We have also compared the steered QAA to a naive classical algorithm that works only for weak interactions. Combining all our results shows that there is a substantial range of parameters for which the steered QAA outperforms both the standard QAA and the naive algorithm.

We are thankful to Sergey Knysh and Vadim Smelyanskiy for fruitful discussions. The simulations were performed using the computing resources and assistance of the UW-Madison Center For High Throughput Computing (CHTC). The work was supported by NSF EAGER Grant No. DMR-1743986.