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Title: Reinforcement Learning via Replica Stacking of Quantum Measurements for the Training of Quantum Boltzmann Machines

Paper content:
Recent theoretical and experimental results suggest the possibility of using current and near-future quantum hardware in challenging sampling tasks.
In this paper, we introduce free-energy-based reinforcement learning (FERL) as an application of quantum hardware.
We propose a method for processing a quantum annealer’s measured qubit spin configurations in approximating the free energy of a quantum Boltzmann machine (QBM).
We then apply this method to perform reinforcement learning on the grid-world problem using the D-Wave 2000Q quantum annealer.
The experimental results show that our technique is a promising method for harnessing the power of quantum sampling in reinforcement learning tasks.
Reinforcement learning (RL) BID33 ; BID6 has been successfully applied in fields such as engineering BID11 ; BID35 , sociology BID12 ; BID30 , and economics BID22 ; BID31 .
The training samples in reinforcement learning are provided by the interaction of an agent with an ambient environment.
For example, in a motion planning problem in uncharted territory, it is desirable for the agent to learn to correctly navigate in the fastest way possible, making the fewest blind decisions.
That is, neither exploration nor exploitation can be pursued exclusively without either facing a penalty or failing at the task.
Our goal is, therefore, not only to design an algorithm that eventually converges to an optimal policy, but for the algorithm to be able to generate suboptimal policies early in the learning process.
Free-energy-based reinforcement learning (FERL) using a restricted Boltzmann machine (RBM), as suggested by BID27 , relies on approximating a utility function for the agent, called the Q-function, using the free energy of an RBM.
RBMs have the advantage that their free energy can be efficiently calculated using closed formulae.
RBMs can represent any joint distribution over binary variables BID20 ; BID15 ; Le BID19 ; however, this property of universality may require exponentially large RBMs BID20 ; Le BID19 .General
Boltzmann machines (GBM) are proposed in an effort to devise universal Q-function approximators with polynomially large Boltzmann networks BID10 . Traditionally
, Monte Carlo simulation is used to perform the computationally expensive tasks of approximating the free energy of GBMs under a Boltzmann distribution. One way to speed
up the approximation process is to represent a GBM by an equivalent physical system and try to find its Boltzmann distribution. An example of such
a physical system is a quantum annealer consisting of a network of pair-wise interacting quantum bits (qubits). Although quantum annealers
have already been used in many areas of computational science, including combinatorial optimization and machine learning, their application in RL has not been explored.In order to use quantum annealing for RL, we first represent the Q-function as the free energy of a physical system, that is, that of a quantum annealer. We then slowly evolve the
state of the physical system from a well-known initial state toward a state with a Boltzmann-like probability distribution. Repeating the annealing process
sufficiently long can provide us with samples from the Boltzmann distribution so that we can empirically approximate the free energy of the physical system under this distribution. Finally, approximating the free
energy of the system would give us an estimate of the Q-function. Up until the past few years, studies
were limited to the classical Boltzmann machines. 1 Recently, BID10 generalized the classical
method toward a quantum or quantum-inspired algorithm for approximating the free energy of GBMs. Using simulated quantum annealing (SQA) BID10
showed that FERL using a deep Boltzmann machine (DBM) can provide a drastic improvement in the early stages of learning, yet performing the same procedure on an actual quantum device remained a difficult task. This is because sampling from a quantum system
representing a quantum Boltzmann machine is harder than the classical case, since at the end of each anneal the quantum system is in a superposition. Any attempt to measure the final state of the
quantum system is doomed to fail since the superposition would collapse into a classical state that does not carry the entirety of information about the final state.In this work, we have two main contributions. We first employ a quantum annealer as a physical
device to approximate the free energy of a classical Boltzmann machine. Second, we generalize the notion of classical Boltzmann
machines to quantum Boltzmann machines within the field of RL and utilize a quantum annealer to approximate the free energy of a quantum system. In order to deal with the issue of superposition mentioned
above, we propose a novel stacking procedure in that we attempt to reconstruct the full state of superposition from the partial information that we get from sampling after each anneal. Finally we report proof-of-concept results using the D-Wave
2000Q quantum processor to provide experimental evidence for the applicability of a quantum annealer in reinforcement learning as predicted by BID10 .
We solve the grid-world problem using various Q-learning methods with the Q-function parametrized by different neural networks.
For comparison, we demonstrate the performance of a fully connected deep Q-network method that can be considered state of the art.
This method efficiently processes every training sample, but, as shown in Fig. 4 , requires a very large number of training samples to converge to the optimal policy.
Another conventional method is free-energy-based RL using an RBM.
This method is also very successful at learning the optimal policy at the scale of the RL task considered in our experiment.
Although this method does not outperform other FERL methods that take advantage of a highly efficient sampling oracle, the processing of each training sample is efficient, as it is based on closed formulae.
In fact, for the size of problem considered, the RBM-based FERL outperforms the fully connected deep Q-network method.The comparison of results in Fig. 6 suggests that replica stacking is a successful method for estimating effective classical configurations obtained from a quantum annealer, given that the spins can only be measured in measurement bases.
For practical use in RL, this method provides a means of treating the quantum annealer as a QBM.
FERL using the quantum annealer, in conjunction with the replica stacking technique, provides significant improvement over FERL using classical Boltzmann machines.
The curve representing SQA-based FERL using a Boltzmann machine on the Chimera graph is almost coincident with the one obtained using the D-Wave 2000Q, whereas the SQA-based FERL using a DBM slightly outperforms it.
This suggests that quantum annealing chips with greater connectivity and more control over annealing time can further improve the performance of the replica stacking method applied to RL tasks.
This is further supported by comparing the performance of SA-based FERL using a DBM versus SA-based FERL using the Chimera graph.
This result shows that DBM is, due to its additional connections, a better choice of neural network compared to the Chimera graph.For practical reasons, we aim to associate an identical choice of virtual parameters β and Γ to all of the TFIMs constructed using FERL.
BID5 and BID25 provide methods for estimating the effective inverse temperature β for other applications.
However, in both studies, the samples obtained from the quantum annealer are matched to the Boltzmann distribution of a classical Ising model.
In fact, the transverse-field strength is a second virtual parameter that we consider.
The optimal choice Γ " 0.5 corresponds to 2{3 of the annealing time, in agreement with the work of BID0 , who also considers TFIM with 16 qubits.The agreement of FERL using quantum annealer reads treated as classical Boltzmann samples with that of FERL using SA and classical Boltzmann machines suggests that, at least for this task and this size of Boltzmann machine, the measurements provided by the D-Wave 2000Q can be considered good approximations of Boltzmann distribution samples of classical Ising models.The extended undirected graphical model developed in this paper using the replica stacking method is not limited to Q-function approximation in RL tasks.
Potentially, this method can be applied to tasks where Boltzmann machines can be used.
This method provides a mechanism for approximating the activations and partition functions of quantum Boltzmann machines that have a significant transverse field.
In this paper, we describe a free-energy-based reinforcement learning algorithm using an existing quantum annealer, namely the D-Wave 2000Q.
Our method relies on the Suzuki-Trotter decomposition and the use of the measured configurations by the D-Wave 2000Q as replicas of an effective classical Ising model of one dimension higher.
The results presented here are first-step proofs of concept of a proposed quantum algorithm with a promising path towards outperforming reinforcement learning algorithms devised for digital hardware.
Given appropriate advances in quantum annealing hardware, future research can employ the proposed principles to solve larger-scale reinforcement learning tasks in the emerging field of quantum machine learning.

assistant: text: We train Quantum Boltzmann Machines using a replica stacking method and a quantum annealer to perform a reinforcement learning task.