Phase-space interference of states optically truncated by quantum scissors: Generation of distinct superpositions of qudit coherent states by displacement of vacuum

Conventional Glauber coherent states (CS) can be defined in several equivalent ways, e.g., by displacing the vacuum or, explicitly, by their infinite Poissonian expansion in Fock states. It is well known that these definitions become inequivalent if applied to finite $d$-level systems (qudits). We present a comparative Wigner-function description of the qudit CS defined (i) by the action of the truncated displacement operator on the vacuum and (ii) by the Poissonian expansion in Fock states of the Glauber CS truncated at $(d-1)$-photon Fock state. These states can be generated from a classical light by its optical truncation using nonlinear and linear quantum scissors devices, respectively. We show a surprising effect that a macroscopically distinguishable superposition of two qudit CS (according to both definitions) can be generated with high fidelity by displacing the vacuum in the qudit Hilbert space. If the qudit dimension $d$ is even (odd) then the superposition state contains Fock states with only odd (even) photon numbers, which can be referred to as the odd (even) qudit CS or the female (male) Schr\"odinger cat state. This phenomenon can be interpreted as an interference of a single CS with its reflection from the highest-energy Fock state of the Hilbert space, as clearly seen via phase-space interference of the Wigner function. We also analyze nonclassical properties of the qudit CS including their photon-number statistics and nonclassical volume of the Wigner function, which is a quantitative parameter of nonclassicality (quantumness) of states. Finally, we study optical tomograms, which can be directly measured in the homodyne detection of the analyzed qudit cat states and enable the complete reconstructions of their Wigner functions.Comment: 14 pages, 10 figure

New class of quantum error-correcting codes for a bosonic mode

We construct a new class of quantum error-correcting codes for a bosonic mode which are advantageous for applications in quantum memories, communication, and scalable computation. These 'binomial quantum codes' are formed from a finite superposition of Fock states weighted with binomial coefficients. The binomial codes can exactly correct errors that are polynomial up to a specific degree in bosonic creation and annihilation operators, including amplitude damping and displacement noise as well as boson addition and dephasing errors. For realistic continuous-time dissipative evolution, the codes can perform approximate quantum error correction to any given order in the timestep between error detection measurements. We present an explicit approximate quantum error recovery operation based on projective measurements and unitary operations. The binomial codes are tailored for detecting boson loss and gain errors by means of measurements of the generalized number parity. We discuss optimization of the binomial codes and demonstrate that by relaxing the parity structure, codes with even lower unrecoverable error rates can be achieved. The binomial codes are related to existing two-mode bosonic codes but offer the advantage of requiring only a single bosonic mode to correct amplitude damping as well as the ability to correct other errors. Our codes are similar in spirit to 'cat codes' based on superpositions of the coherent states, but offer several advantages such as smaller mean number, exact rather than approximate orthonormality of the code words, and an explicit unitary operation for repumping energy into the bosonic mode. The binomial quantum codes are realizable with current superconducting circuit technology and they should prove useful in other quantum technologies, including bosonic quantum memories, photonic quantum communication, and optical-to-microwave up- and down-conversion.Comment: Published versio

Three Results in Quantum Physics

This thesis is split into three disjoint sections. The first deals with two practical issues regarding the use of unitary 2-designs. A simplified description of how to generate elements of the smallest known unitary 2-design on qubits is given which should be usable even for people who do not have much experience with the mathematics of finite fields. The section also gives a new way to decompose an arbitrary element of the Clifford group into one and two qubit gates and is by far the simplest decomposition of its kind. The second section describes similarities and differences between a probabilistic formulation of classical mechanics and quantum mechanics, with the intention that it could become a resource for physics students to show that just because a physical phenomenon is strange it is not necessarily quantum. The third section is speculative and delves into the relationship between a highly theoretical field of quantum information science, Quantum Prover Interactive Proofs, and a highly practical area of quantum information science, error characterization. Previously unnoticed links are drawn between these fields with the intention that further research can provide fertile ground for both to flourish

Single-step transfer or exchange of multipartite quantum entanglement with minimum resources

The transfer or exchange of multipartite quantum states is critical to the realization of large-scale quantum information processing and quantum communication. A challenging question in this context is: What is the minimum resource required and how to simultaneously transfer or exchange multipartite quantum entanglement between two sets of qubits. Finding the answer to these questions is of great importance to quantum information science. In this work, we demonstrate that by using a single quantum two-level system - the simplest quantum object - as a coupler arbitrary multipartite quantum states (either entangled or separable) can be transferred or exchanged simultaneously between two sets of qubits. Our findings offer the potential to significantly reduce the resources needed to construct and operate large-scale quantum information networks consisting of many multi-qubit registers, memory cells, and processing units.Comment: 45 pages, 7 figure

Parallelizing quantum circuit synthesis

We present an algorithmic framework for parallel quantum circuit synthesis using meet-in-the-middle synthesis techniques. We also present two implementations thereof, using both threaded and hybrid parallelization techniques. We give examples where applying parallelism offers a speedup on the time of circuit synthesis for 2- and 3-qubit circuits. We use a threaded algorithm to synthesize 3-qubit circuits with optimal T -count 9, and 11, breaking the previous record of T-count 7. As the estimated runtime of the framework is inversely proportional to the number of processors, we propose an implementation using hybrid parallel programming which can take full advantage of a computing cluster’s thousands of cores. This implementation has the potential to synthesize circuits which were previously deemed impossible due to the exponential runtime of existing algorithms

Resource optimization for fault-tolerant quantum computing

In this thesis we examine a variety of techniques for reducing the resources required for fault-tolerant quantum computation. First, we show how to simplify universal encoded computation by using only transversal gates and standard error correction procedures, circumventing existing no-go theorems. We then show how to simplify ancilla preparation, reducing the cost of error correction by more than a factor of four. Using this optimized ancilla preparation, we develop improved techniques for proving rigorous lower bounds on the noise threshold. Additional overhead can be incurred because quantum algorithms must be translated into sequences of gates that are actually available in the quantum computer. In particular, arbitrary single-qubit rotations must be decomposed into a discrete set of fault-tolerant gates. We find that by using a special class of non-deterministic circuits, the cost of decomposition can be reduced by as much as a factor of four over state-of-the-art techniques, which typically use deterministic circuits. Finally, we examine global optimization of fault-tolerant quantum circuits under physical connectivity constraints. We adapt techniques from VLSI in order to minimize time and space usage for computations in the surface code, and we develop a software prototype to demonstrate the potential savings.Comment: 231 pages, Ph.D. thesis, University of Waterlo

Topological Code Architectures for Quantum Computation

This dissertation is concerned with quantum computation using many-body quantum systems encoded in topological codes. The interest in these topological systems has increased in recent years as devices in the lab begin to reach the fidelities required for performing arbitrarily long quantum algorithms. The most well-studied system, Kitaev\u27s toric code, provides both a physical substrate for performing universal fault-tolerant quantum computations and a useful pedagogical tool for explaining the way other topological codes work. In this dissertation, I first review the necessary formalism for quantum information and quantum stabilizer codes, and then I introduce two families of topological codes: Kitaev\u27s toric code and Bombin\u27s color codes. I then present three chapters of original work. First, I explore the distinctness of encoding schemes in the color codes. Second, I introduce a model of quantum computation based on the toric code that uses adiabatic interpolations between static Hamiltonians with gaps constant in the system size. Lastly, I describe novel state distillation protocols that are naturally suited for topological architectures and show that they provide resource savings in terms of the number of required ancilla states when compared to more traditional approaches to quantum gate approximation