Abstract Quantum simulation is one of the central discipline to demonstrate the power of quantum computing. In recent years, the theoretical framework of quantum superchannels has been developed and applied widely as the extension of quantum channels. In this work, we study the quantum circuit simulation task of superchannels. We develop a quantum superchannel simulation algorithm based on the convex decomposition into sum of extreme superchannels. We demonstrate the algorithm by numerical simulation of qubit superchannels with high accuracy, making it applicable to current experimental platforms. Our study stands as an expansion of the superchannel theory to the field of quantum simulation and algorithm, as well as an extension of quantum simulation from channels and open-system dynamics to superchannels and processes with manifest quantum memory effects.
Abstract Quantum simulation is one of the central discipline to demonstrate the power of quantum computing. In recent years, the theoretical framework of quantum superchannels has been developed and applied widely as the extension of quantum channels. In this work, we study the quantum circuit simulation task of superchannels. We develop a quantum superchannel simulation algorithm based on the convex decomposition into sum of extreme superchannels. We demonstrate the algorithm by numerical simulation of qubit superchannels with high accuracy, making it applicable to current experimental platforms. Our study stands as an expansion of the superchannel theory to the field of quantum simulation and algorithm, as well as an extension of quantum simulation from channels and open-system dynamics to superchannels and processes with manifest quantum memory effects.
Quantum simulation, which is generically the task to employ quantum computers to simulate quantum physical models, has been one of the most significant motivations of quantum computing. Quantum dynamics, unitary or nonunitary Markovian dynamics driven by local interactions, has been proved to be efficiently simulatable on quantum computers. Extending the underlying models from unitary to nonunitary evolution, and from continuous-time to discrete-time evolution is essential not only for quantum simulation of more general processes, e.g., dissipative processes with evident non-Markovian effects, but also for developing alternative quantum computing models and algorithms. In this thesis, we explore quantum simulation problems mainly from the following three themes. First, we extend quantum simulation framework of Hamiltonian-driven evolution to quantum simulation of quantum channels, combined with the scheme of algorithmic simulation that accepts a promised simulation accuracy, hence algorithmic quantum channel simulation. Our simulation scheme contains a classical preprocessing part, i.e. a classical algorithm for quantum-circuit design, and a quantum part that is the execution of a quantum circuit for the quantum channel simulator. Second, we employ simulation algorithms for arbitrary quantum channels beyond the dilation method. We explore channel decomposition in terms of convex combination of smaller channels, known as generalized extreme channels, which is known as a nontrivial open problem. To attack this problem, we develop an optimization algorithm for approximate decomposition into a convex sum of generalized extreme channels. We provide an ansatz that proves to be able to yield arbitrary generalized extreme channels and allows a precise quantum circuit description. Furthermore, our numerical simulation has demonstrated the validity of our optimization algorithm for low-dimensional quantum channels. Third, by considering quantum simulation problems beyond the rough distinction between digital and analog simulations, and beyond the quantum-state generation problem, we define uniform, strong, and weak quantum simulations from the point view of operator topology. Besides strong simulation of quantum channels, we define a general weak quantum simulation problem, which simulates observable effects instead of the effects on state generation. Also we study the channel simulation problem in the quantum query model, and provide the query complexity by employing uniform quantum simulation method.
Abstract Simulating quantum physical processes has been one of the major motivations for quantum information science. Quantum channels, which are completely positive and trace preserving processes, are the standard mathematical language to describe quantum evolution, while in recent years quantum superchannels have emerged as the substantial extension. Superchannels capture effects of quantum memory and non-Markovianality more precisely, and have found broad applications in universal models, algorithm, metrology, discrimination tasks, as examples. Here, we report an experimental simulation of qubit superchannels in a nuclear magnetic resonance (NMR) system with high accuracy, based on a recently developed quantum algorithm for superchannel simulation. Our algorithm applies to arbitrary target superchannels, and our experiment shows the high quality of NMR simulators for near-term usage. Our approach can also be adapted to other experimental systems and demonstrates prospects for more applications of superchannels.
Quantum channels, which are completely positive and trace preserving mappings, can alter the dimension of a system, e.g. a quantum channel from a qubit to a qutrit. We study the convex set properties of dimension-altering quantum channels, and particularly the channel decomposition problem in terms of convex sum of extreme channels. We provide various quantum circuit representations of extreme and generalized extreme channels, which can be employed in an optimization to approximately decompose an arbitrary channel. Numerical simulations of low-dimensional channels are performed to demonstrate our channel decomposition scheme.
Quantum channels describe general transformations of the state of a physical system and play a fundamental role in many fields. It promises wide range of applications, because any physical process can be represented as a quantum channel transforming an initial state into a final state. The evolution of system undergoing a quantum channel is not necessary unitary which makes it unrealisable without ancillary system. Inspired by the method performing non-unitary operator by the linear combination of unitary operations, we proposed a quantum algorithm for the simulation of universal single-qubit channel, described by a convex combination of two generalized extreme channels corresponding to four Kraus operators, and is scalable to arbitrary higher dimension. We demonstrate the whole algorithm experimentally using the universal IBM cloud quantum computer and study properties of different qubit quantum channels. We illustrate the quantum capacity of the general qubit quantum channels, which quantifies the amount of quantum information that can be protected. There is a general agreement between the theoretical predictions and the experiments, which strongly supported our method. By realizing arbitrary qubit channel, this work provides a universal way to explore various properties of quantum channel and novel prospect of quantum communication.
A quantum algorithm is presented for the simulation of arbitrary Markovian dynamics of a qubit, described by a semigroup of single-qubit quantum channels Tt specified by a generator L. This algorithm requires only single-qubit and controlled-not gates and approximates the channel Tt=etL up to the chosen accuracy $\ensuremath{\epsilon}$, with a slightly superlinear cost ${O((\ensuremath{\parallel}\mathcal{L}\ensuremath{\parallel}}_{(1\ensuremath{\rightarrow}1)}t{)}^{1+1/2k}/{\ensuremath{\epsilon}}^{1/2k})$ for any integer k. Inspired by developments in Hamiltonian simulation, a decomposition and recombination technique is utilized which allows for the exploitation of recently developed methods for the approximation of arbitrary single-qubit channels. In particular, as a result of these methods the algorithm requires only a single ancilla qubit, the minimal possible dilation for a nonunitary single-qubit quantum channel.
Recently quantum computing has achieved significant results both in theoretical and experimental areas and quantum algorithms have drawn the attention of many scientists. At the present there exist several experiments that realized some parts of certain important quantum algorithms. While those experiments work with only up to seven qubits, quantum simulators allow scientists to design algorithms with many more qubits. The existing simulators have one or more of the following restrictions: either they require powerful computers and additional computation packages such as Matlab or Mathematica or they don't provide an easy method for interacting between the classical and quantum parts of an algorithm. More importantly, none of them can simulate a generic 64-qubit circuit. It is important at this stage of quantum computing development to investigate technological improvements in simulating quantum circuit design.
In this thesis we developed a specialized data structure, QSV, for quantum simulators along with new method for calculation of the new state vector after application of a quantum gate in a quantum simulator. We also developed a technique that speeds up the simulation time with our QSV vectors. With those results we built a quantum simulator that overcomes many shortcomings in other simulators and it has many advanced features.
Our simulator can simulate many 64-qubit circuits and it can directly simulate multiple qubit gates as well as allows users to define gates with up to 10 qubits. The simulation time is much less in comparison with other simulators we tested. The simulator defines specifications so that users can easily design and test quantum circuits. With the simulator we also developed some new quantum circuits and verified many designs and constructions.