Framework for universal NMR quantum computing using Heisenberg spin interaction

Quantum computing using the control techniques of nuclear magnetic resonance (NMR) has been one of the first experimental implementations of quantum informa- tion processing. By rotating nuclear spins inside molecules with magnetic fields, it is possible to implement any unitary operation on a set of spin-1/2-qubits. Since published work has so far been limited to the Ising spin interaction, this thesis extends the framework of NMR quantum computing to the Heisenberg interaction. In order to find NMR pulse sequences that represent quantum gates in the machine language of NMR quantum computing, magnetic and radio-frequency fields, an algorithm was implemented to examine billions of possible sequences for a universal set of quantum gates. The Python program was optimized to avoid the numerically most expensive calculations so that sequences up to nine pulses could be investigated. The search yielded an NMR pulse sequence for the anisotropic Heisenberg interaction that im- plements the CNOT quantum gate on an arbitrary input state. However, no such sequence was found for the isotropic Heisenberg interaction for a sequence of up to nine pulses in length and while restricting the single-qubit rotations to a finite set of rotation angles. The framework of NMR quantum computing can thus be extended to the Heisenberg interaction although it is not clear if universal quantum computing is possible using only the isotropic Heisenberg interaction to entangle two qubits

Faster ground state energy estimation on early fault-tolerant quantum computers via rejection sampling

A major thrust in quantum algorithm development over the past decade has been the search for the quantum algorithms that will deliver practical quantum advantage first. Today's quantum computers and even early fault-tolerant quantum computers will be limited in the number of operations they can implement per circuit. We introduce quantum algorithms for ground state energy estimation (GSEE) that accommodate this design constraint. The first estimates ground state energies and has a quadratic improvement on the ground state overlap parameter compared to other methods in this regime. The second certifies that the estimated ground state energy is within a specified error tolerance of the true ground state energy, addressing the issue of gap estimation that beleaguers several ground state preparation and energy estimation algorithms. We note, however, that the scaling of this certification technique is, unfortunately, worse than that of the GSEE algorithm. These algorithms are based on a novel use of the quantum computer to facilitate rejection sampling. After a classical computer is used to draw samples, the quantum computer is used to accept or reject the samples. The set of accepted samples correspond to draws from a target distribution. While we use this technique for ground state energy estimation, it may find broader application. Our work pushes the boundaries of what operation-limited quantum computers are capable of and thus brings the target of quantum advantage closer to the present.Comment: 31 pages + appendix, 5 figure

A quantum genetic algorithm with quantum crossover and mutation operations

In the context of evolutionary quantum computing in the literal meaning, a quantum crossover operation has not been introduced so far. Here, we introduce a novel quantum genetic algorithm which has a quantum crossover procedure performing crossovers among all chromosomes in parallel for each generation. A complexity analysis shows that a quadratic speedup is achieved over its classical counterpart in the dominant factor of the run time to handle each generation.Comment: 21 pages, 1 table, v2: typos corrected, minor modifications in sections 3.5 and 4, v3: minor revision, title changed (original title: Semiclassical genetic algorithm with quantum crossover and mutation operations), v4: minor revision, v5: minor grammatical corrections, to appear in QI

Algorithms on ensemble quantum computers.

In ensemble (or bulk) quantum computation, all computations are performed on an ensemble of computers rather than on a single computer. Measurements of qubits in an individual computer cannot be performed; instead, only expectation values (over the complete ensemble of computers) can be measured. As a result of this limitation on the model of computation, many algorithms cannot be processed directly on such computers, and must be modified, as the common strategy of delaying the measurements usually does not resolve this ensemble-measurement problem. Here we present several new strategies for resolving this problem. Based on these strategies we provide new versions of some of the most important quantum algorithms, versions that are suitable for implementing on ensemble quantum computers, e.g., on liquid NMR quantum computers. These algorithms are Shor's factorization algorithm, Grover's search algorithm (with several marked items), and an algorithm for quantum fault-tolerant computation. The first two algorithms are simply modified using a randomizing and a sorting strategies. For the last algorithm, we develop a classical-quantum hybrid strategy for removing measurements. We use it to present a novel quantum fault-tolerant scheme. More explicitly, we present schemes for fault-tolerant measurement-free implementation of Toffoli and σ(z)(¼) as these operations cannot be implemented "bitwise", and their standard fault-tolerant implementations require measurement