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Register a new userQuantum Logic Gate Synthesis as a Markov Decision Process
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Authors
M. Sohaib Alam
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Reinforcement learning has witnessed recent applications to a variety of tasks in quantum programming. The underlying assumption is that those tasks could be modeled as Markov Decision Processes (MDPs). Here, we investigate the feasibility of this assumption by exploring its consequences for two of the simplest tasks in quantum programming: state preparation and gate compilation. By forming discrete MDPs, focusing exclusively on the single-qubit case, we solve for the optimal policy exactly through policy iteration. We find optimal paths that correspond to the shortest possible sequence of gates to prepare a state, or compile a gate, up to some target accuracy. As an example, we find sequences of H and T gates with length as small as 11 producing ~99% fidelity for states of the form (HT)^{n} |0> with values as large as n=10^{10}. This work provides strong evidence that reinforcement learning can be used for optimal state preparation and gate compilation for larger qubit spaces.
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We present a synthesis framework to map logic networks into quantum circuits for quantum computing. The synthesis framework is based on LUT networks (lookup-table networks), which play a key role in conventional logic synthesis. Establishing a connection between LUTs in a LUT network and reversible single-target gates in a reversible network allows us to bridge conventional logic synthesis with logic synthesis for quantum computing, despite several fundamental differences. We call our synthesis framework LUT-based Hierarchical Reversible Logic Synthesis (LHRS). Input to LHRS is a classical logic network; output is a quantum network (realized in terms of Clifford+$T$ gates). The framework offers to trade-off the number of qubits for the number of quantum gates. In a first step, an initial network is derived that only consists of single-target gates and already completely determines the number of qubits in the final quantum network. Different methods are then used to map each single-target gate into Clifford+$T$ gates, while aiming at optimally using available resources. We demonstrate the effectiveness of our method in automatically synthesizing IEEE compliant floating point networks up to double precision. As many quantum algorithms target scientific simulation applications, they can make rich use of floating point arithmetic components. But due to the lack of quantum circuit descriptions for those components, it can be difficult to find a realistic cost estimation for the algorithms. Our synthesized benchmarks provide cost estimates that allow quantum algorithm designers to provide the first complete cost estimates for a host of quantum algorithms. Thus, the benchmarks and, more generally, the LHRS framework are an essential step towards the goal of understanding which quantum algorithms will be practical in the first generations of quantum computers.
We study the synthesis of mode switching protocols for a class of discrete-time switched linear systems in which the mode jumps are governed by Markov decision processes (MDPs). We call such systems MDP-JLS for brevity. Each state of the MDP corresponds to a mode in the switched system. The probabilistic state transitions in the MDP represent the mode transitions. We focus on finding a policy that selects the switching actions at each mode such that the switched system that follows these actions is guaranteed to be stable. Given a policy in the MDP, the considered MDP-JLS reduces to a Markov jump linear system (MJLS). {We consider both mean-square stability and stability with probability one. For mean-square stability, we leverage existing stability conditions for MJLSs and propose efficient semidefinite programming formulations to find a stabilizing policy in the MDP. For stability with probability one, we derive new sufficient conditions and compute a stabilizing policy using linear programming. We also extend the policy synthesis results to MDP-JLS with uncertain mode transition probabilities.
The Markov assumption (MA) is fundamental to the empirical validity of reinforcement learning. In this paper, we propose a novel Forward-Backward Learning procedure to test MA in sequential decision making. The proposed test does not assume any parametric form on the joint distribution of the observed data and plays an important role for identifying the optimal policy in high-order Markov decision processes and partially observable MDPs. We apply our test to both synthetic datasets and a real data example from mobile health studies to illustrate its usefulness.
To demonstrate the advantage of quantum computation, many attempts have been made to attack classically intractable problems, such as the satisfiability problem (SAT), with quantum computer. To use quantum algorithms to solve these NP-hard problems, a quantum oracle with quantum circuit implementation is usually required. In this manuscript, we first introduce a novel algorithm to synthesize quantum logic in the Conjunctive Normal Form (CNF) model. Compared with linear ancillary qubits in the implementation of Qiskit open-source framework, our algorithm can synthesize an m clauses n variables k-CNF with $O(k^2 m^2/n)$ quantum gates by only using three ancillary qubits. Both the size and depth of the circuit can be further compressed with the increase in the number of ancillary qubits. When the number of ancillary qubits is $Omega(m^epsilon)$ (for any $epsilon > 0$), the size of the quantum circuit given by the algorithm is O(km), which is asymptotically optimal. Furthermore, we design another algorithm to optimize the depth of the quantum circuit with only a small increase in the size of the quantum circuit. Experiments show that our algorithms have great improvement in size and depth compared with the previous algorithms.
Integrated quantum photonics provides a promising route towards scalable solid-state implementations of quantum networks, quantum computers, and ultra-low power opto-electronic devices. A key component for many of these applications is the photonic quantum logic gate, where the quantum state of a solid-state quantum bit (qubit) conditionally controls the state of a photonic qubit. These gates are crucial for development of robust quantum networks, non-destructive quantum measurements, and strong photon-photon interactions. Here we experimentally realize a quantum logic gate between an optical photon and a solid-state qubit. The qubit is composed of a quantum dot (QD) strongly coupled to a nano-cavity, which acts as a coherently controllable qubit system that conditionally flips the polarization of a photon on picosecond timescales, implementing a controlled-NOT (cNOT) gate. Our results represent an important step towards solid-state quantum networks and provide a versatile approach for probing QD-photon interactions on ultra-fast timescales.
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