Benchmark Results

In order to validate the simulator, we performed some initial tests using basic quantum circuits such as all Hadamard gates (on each qubit) and the Quantum Fourier Transform (QFT). The main purpose of those tests is to confirm the threading scalability of the path integral approach, i.e. we get a linear reduction in runtime when more computing resources (CPU cores) are added. This is the main difference between the path integral and the linear-algebra methods where the latter is often constrained by memory traffic.

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First, we ran a scalability test in which we gradually increased the number of computing threads available for the simulator. The data for a test suite consists of all Hadamard gates on a 30-qubit register is shown in Fig. 5.7.

Fig. 5.7 Run-time vs. number of computing thread benchmark of the simulator running all Hadamard gates on a 30-qubit register.

Fig. 5.7 clearly demonstrates the linear scaling that we expect from a path-integral simulator, especially when the thread count is in the range from 16 to 256 threads. Outside that range of core count, there are a few hardware-related effects kick in which changes the expected scaling behaviours. In the low core count range (8 vs. 16), since the number of cores per CPU socket is 24, running only a few cores, e.g. 8 cores, will allow the CPU to throttle the clock frequency of those executing cores hence the difference in runtime is minimal when we double the core count in this case. On the other hand, the total number of physical hardware cores that are available is 384. Hence, using more threads than that will not result in any significant performance improvement as we can see in the tail-end of Fig. 5.7.

Next, we compare the ATOS path-integral simulator against the publicly available Microsoft LIQU i_|⟩(Fig. 5.8) and a generic linear algebra simulator such as Intel qHiP- STER (Fig. 5.9). In both cases, we use Quantum Fourier Transform as the test case.

130 Chapter 5. Quantum Programming and Simulation

Fig. 5.8 Run-time comparison between the ATOS simulator and LIQU i_|⟩running the Quantum Fourier Transform using the same number of threads (128). The initial state is

|00...0_⟩.

Fig. 5.9 Run-time comparison between the simulator and linear algebra (LA) based simulator. We implemented a generic LA quantum simulator using the same computing model as qHiPSTER so that we can run on the local Bullion server. The runtime data of the LA simulator matches the public data of qHiPSTER (after compensating the memory bandwidth differences) as shown by the red triangles in the cases of 29 and 30 qubits. The ATOS path-integral simulator provides faster time-to-result in this case as compared to the LA-based simulator.

When comparing to Microsoft LIQU i_|⟩, Fig. 5.8 shows a significant performance gap in favour of the ATOS simulator. This can be explained partly because the ATOS simulator is heavily optimised for this particular hardware configuration while the LIQU i_|⟩simulator is designed mainly for general PC-based setups. Secondly, the path-

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integral based simulator can take advantage of the fact that the input state is classical hence eliminating a lot of configuration paths.

Fig. 5.9, on the other hand, shows a much closer gap between an optimised LA-based simulator and a path integral based counterpart. The path-integral simulator is still faster in this case stemming from the fact that the input state is singular. One thing that we can observe in Fig. 5.9 is the almost doubling of run time for the path integral simulator when a qubit is added. This is due to the fact that the number of paths which need to be explored doubles.

Preliminary data from the ATOS quantum simulator has been promising thanks to its innovative approach to maximising low-level parallelism to achieve speed-up in various use cases. The performance advantage is also derived from software and hardware co-design for the high-end Bull Bullion HPC platforms. A proposal to study the feasibility of using FPGA as an accelerator for this type of simulation is summarized in Appendix B. The simulator is further enhanced and maintained by the ATOS Quantum Lab after the author finished the research internship there.

Chapter 6

Simulating Input-Output Quantum

Systems with LIQU i|⟩

Never try to solve all the problems at once — make them line up for you one-by-one.

Richard Sloma

Quantum computation is usually described by unitary operations which is based on closed quantum system dynamics. In contrast,open quantum dynamical systems are often modeled by the master equation which takes into account the environmental interaction. Using the complementary Heisenberg-picture approach, we can derive the quantum stochastic differential equations that describe the dynamics of system operators of an open quantum plant. This is commonly referred to as the quantum input-output formalism, which underpins the burgeoning field of quantum feedback control.

The inclusion of dissipation processes, allows us to passively and autonomously perform a wide variety of quantum operations, such as enhancing adiabatic quantum computation, and state preparation/stabilization. For instance, Chapter 3 and 4 have demonstrated the application of open quantum system models in developing protocols to improve the stability of qubit systems. More importantly, fully coherent feedback loops mediated by field coupling without measurement have been shown to have major advantages such as high operating speed for quantum optics applications and the non- demolition nature of coherent operations.

134 Chapter 6. Simulating Input-Output Quantum Systems with LIQU i_|⟩

In this chapter, we describe the use of Microsoft Language-Integrated Quantum Operations framework (LIQU i_|⟩[194]) in simulating complex engineeredopenquantum systems by discretizing the physical system composition, and introducing a dissipation emulator gadget. The Hamiltonian simulator module of LIQU i_|⟩ is used for time evolution simulation of the Trotter decomposed unitary.

In particular, we will apply this method to compound gradient echo quantum memory which is capable of efficiently capturing and storing a quantum state of light. In these experiments, memory hardware consists of a continuous solid-state material (rare-earth- ion-doped crystal) which will be represented in simulation by a network of qubits. This not only provides analysis results for such a discrete quantum memory model, but can also potentially be used as a blueprint for a quantum digital signal processor capable of processing incoming continuous variable quantum information.