Random unitary matrices sampled from the uniform Haar ensemble have a number of important applications both in cryptography and in the simulation of a variety of fundamental physical systems. Since the Haar ensemble is very expensive to sample, pseudorandom ensembles in the form of t-designs are frequently used as an efficient substitute, and are sufficient for most applications. We investigate t-designs generated using a measurement-based approach on superconducting quantum computers. In particular, we implemented an exact single-qubit 3-design on IBM quantum processors by performing measurements on a 6-qubit graph state. By analysing channel tomography results, we were able to show that the ensemble of unitaries realised was a 1-design, but not a 2-design or a 3-design under the test conditions set, which we show to be a result of depolarising noise during the measurement-based process. We obtained improved results for the 2-design test by implementing an approximate 2-design, in which measurements were performed on a smaller 5-qubit graph state, but the test still did not pass for all states. This suggests that the practical realisation of measurement-based t-designs on superconducting quantum computers will require further work on the reduction of depolarising noise in these devices.
Random unitary matrices sampled from the uniform Haar ensemble have a number of important applications both in cryptography and in the simulation of a variety of fundamental physical systems. Since the Haar ensemble is very expensive to sample, pseudorandom ensembles in the form of t-designs are frequently used as an efficient substitute, and are sufficient for most applications. We investigate t-designs generated using a measurement-based approach on superconducting quantum computers. In particular, we implemented an exact single-qubit 3-design on IBM quantum processors by performing measurements on a 6-qubit graph state. By analysing channel tomography results, we were able to show that the ensemble of unitaries realised was a 1-design, but not a 2-design or a 3-design under the test conditions set, which we show to be a result of depolarising noise during the measurement-based process. We obtained improved results for the 2-design test by implementing an approximate 2-design, in which measurements were performed on a smaller 5-qubit graph state, but the test still did not pass for all states. This suggests that the practical realisation of measurement-based t-designs on superconducting quantum computers will require further work on the reduction of depolarising noise in these devices.
This thesis is focused on the generation and understanding of particular kinds of quantum randomness. Randomness is useful for many tasks in physics and information processing, from randomized benchmarking , to black hole physics , as well demonstrating a so-called quantum speedup , and many other applications. On the one hand we explore how to generate a particular form of random evolution known as a t-design. On the other we show how this can also give instances for quantum speedup - where classical computers cannot simulate the randomness efficiently. We also show that this is still possible in noisy realistic settings. More specifically, this thesis is centered around three main topics. The first of these being the generation of epsilon-approximate unitary t-designs. In this direction, we first show that non-adaptive, fixed measurements on a graph state composed of poly(n,t,log(1/epsilon)) qubits, and with a regular structure (that of a brickwork state) effectively give rise to a random unitary ensemble which is a epsilon-approximate t-design. This work is presented in Chapter 3. Before this work, it was known that non-adaptive fixed XY measurements on a graph state give rise to unitary t-designs , however the graph states used there were of complicated structure and were therefore not natural candidates for measurement based quantum computing (MBQC), and the circuits to make them were complicated. The novelty in our work is showing that t-designs can be generated by fixed, non-adaptive measurements on graph states whose underlying graphs are regular 2D lattices. These graph states are universal resources for MBQC. Therefore, our result allows the natural integration of unitary t-designs, which provide a notion of quantum pseudorandomness which is very useful in quantum algorithms, into quantum algorithms running in MBQC. Moreover, in the circuit picture this construction for t-designs may be viewed as a constant depth quantum circuit, albeit with a polynomial number of ancillas. We then provide new constructions of epsilon-approximate unitary t-designs both in the circuit model and in MBQC which are based on a relaxation of technical requirements in previous constructions. These constructions are found in Chapters 4 and 5.
Abstract Quantum computers have the potential to outperform classical computers in a range of computational tasks, such as prime factorisation and unstructured searching. However, real-world quantum computers are subject to noise. Quantifying noise is of vital importance, since it is often the dominant factor preventing the successful realisation of advanced quantum computations. Here we propose and demonstrate an interleaved randomised benchmarking protocol for measurement-based quantum computers that can be used to estimate the fidelity of any single-qubit measurement-based gate. We tested the protocol on IBM superconducting quantum processors by estimating the fidelity of the Hadamard and T gates—a universal single-qubit gate set. Measurements were performed on entangled cluster states of up to 31 qubits. Our estimated gate fidelities show good agreement with those calculated from quantum process tomography. By artificially increasing noise, we were able to show that our protocol detects large noise variations in different implementations of a gate.
Entangled multipartite states are resources for universal quantum computation, but they can also give rise to ensembles of unitary transformations, a topic usually studied in the context of random quantum circuits. Using several graph state techniques, we show that these resources can "derandomize" circuit results by sampling the same kinds of ensembles quantum mechanically, analogously to a quantum random number generator. Furthermore, we find simple examples that give rise to new ensembles whose statistical moments exactly match those of the uniformly random distribution over all unitaries up to order t, while foregoing adaptive feedforward entirely. Such ensembles-known as t designs-often cannot be distinguished from the "truly" random ensemble, and so they find use in many applications that require this implied notion of pseudorandomness.
Measurement based (MB) quantum computation allows for universal quantum computing by measuring individual qubits prepared in entangled multipartite states, known as graph states. Unless corrected for, the randomness of the measurements leads to the generation of ensembles of random unitaries, where each random unitary is identified with a string of possible measurement results. We show that repeating an MB scheme an efficient number of times, on a simple graph state, with measurements at fixed angles and no feedforward corrections, produces a random unitary ensemble that is an $\ensuremath{\varepsilon}$-approximate t design on n qubits. Unlike previous constructions, the graph is regular and is also a universal resource for measurement based quantum computing, closely related to the brickwork state.
At its core a t-design is a method for sampling from a set of unitaries in a way which mimics sampling randomly from the Haar measure on the unitary group, with applications across quantum information processing and physics. We construct new families of quantum circuits on n-qubits giving rise to ε-approximate unitary t-designs efficiently in O(n3t12) depth. These quantum circuits are based on a relaxation of technical requirements in previous constructions. In particular, the construction of circuits which give efficient approximate t-designs by Brandao, Harrow, and Horodecki (F.G.S.L Brandao, A.W Harrow, and M. Horodecki, Commun. Math. Phys. (2016).) required choosing gates from ensembles which contained inverses for all elements, and that the entries of the unitaries are algebraic. We reduce these requirements, to sets that contain elements without inverses in the set, and non-algebraic entries, which we dub partially invertible universal sets. We then adapt this circuit construction to the framework of measurement based quantum computation(MBQC) and give new explicit examples of n-qubit graph states with fixed assignments of measurements (graph gadgets) giving rise to unitary t-designs based on partially invertible universal sets, in a natural way. We further show that these graph gadgets demonstrate a quantum speedup, up to standard complexity theoretic conjectures. We provide numerical and analytical evidence that almost any assignment of fixed measurement angles on an n-qubit cluster state give efficient t-designs and demonstrate a quantum speedup.
We develop the concept of a unitary t-design as a means of expressing operationally useful subsets of the stochastic properties of the uniform (Haar) measure on the unitary group U(2n) on n qubits. In particular, sets of unitaries forming 2-designs have wide applicability to quantum information protocols. We devise an O(n)-size in-place circuit construction for an approximate unitary 2-design. We then show that this can be used to construct an efficient protocol for experimentally characterizing the fidelity of a quantum process on n qubits with quantum circuits of size O(n) without requiring any ancilla qubits, thereby improving upon previous approaches.