Restricted models of quantum computation are ones that have less power than a universal quantum computer. We studied the consequences of removing particular properties from a universal quantum computer to discover whether those resources were important.In the first part of the thesis we studied universal quantum computers which are implemented using Clifford gates, adaptive measurements, and magic states. The Gottesman–Knill theorem shows that circuits in this form which do not use magic states can be simulated by a classical computer. We extended this result to show that all circuits in this form can be partially simulated; the same computation can be implemented using a smaller quantum computer with the assistance of some polynomial time classical computation. We also identified a subclass of these computations that can be shown to not be entirely classically simulated by any method, given certain complexity theoretic assumptions are true.In the next part of the thesis we examine the role of entanglement in noisy quantum computations. Entanglement is necessary for noiseless quantum computers to have any quantum advantage, but it is not known whether the same is true for mixed state quantum computers. We show that entanglement, unexpectedly, does play a crucial role in the most well known mixed state computer: the one clean qubit model.Finally, we investigate how closely classical simulation is related to another idea of classicality.This notion captures how easily the final state of a computation can be learnt, given samples of measurements from it. We find an extra condition under which a circuit that is classically simulable is also efficiently learnable.
Restricted models of quantum computation are ones that have less power than a universal quantum computer. We studied the consequences of removing particular properties from a universal quantum computer to discover whether those resources were important.In the first part of the thesis we studied universal quantum computers which are implemented using Clifford gates, adaptive measurements, and magic states. The Gottesman–Knill theorem shows that circuits in this form which do not use magic states can be simulated by a classical computer. We extended this result to show that all circuits in this form can be partially simulated; the same computation can be implemented using a smaller quantum computer with the assistance of some polynomial time classical computation. We also identified a subclass of these computations that can be shown to not be entirely classically simulated by any method, given certain complexity theoretic assumptions are true.In the next part of the thesis we examine the role of entanglement in noisy quantum computations. Entanglement is necessary for noiseless quantum computers to have any quantum advantage, but it is not known whether the same is true for mixed state quantum computers. We show that entanglement, unexpectedly, does play a crucial role in the most well known mixed state computer: the one clean qubit model.Finally, we investigate how closely classical simulation is related to another idea of classicality.This notion captures how easily the final state of a computation can be learnt, given samples of measurements from it. We find an extra condition under which a circuit that is classically simulable is also efficiently learnable.
We study the computational power of unitary Clifford circuits with solely magic state inputs (CM circuits), supplemented by classical efficient computation. We show that CM circuits are hard to classically simulate up to multiplicative error (assuming polynomial hierarchy non-collapse), and also up to additive error under plausible average-case hardness conjectures. Unlike other such known classes, a broad variety of possible conjectures apply. Along the way, we give an extension of the Gottesman-Knill theorem that applies to universal computation, showing that for Clifford circuits with joint stabilizer and non-stabilizer inputs, the stabilizer part can be eliminated in favour of classical simulation, leaving a Clifford circuit on only the non-stabilizer part. Finally, we discuss implementational advantages of CM circuits.
We study the computational power of unitary Clifford circuits with solely magic state inputs (CM circuits), supplemented by classical efficient computation. We show that CM circuits are hard to classically simulate up to multiplicative error (assuming polynomial hierarchy non-collapse), and also up to additive error under plausible average-case hardness conjectures. Unlike other such known classes, a broad variety of possible conjectures apply. Along the way, we give an extension of the Gottesman-Knill theorem that applies to universal computation, showing that for Clifford circuits with joint stabilizer and non-stabilizer inputs, the stabilizer part can be eliminated in favour of classical simulation, leaving a Clifford circuit on only the non-stabilizer part. Finally, we discuss implementational advantages of CM circuits.
The stabiliser formalism is a widely used and successful subtheory of quantum mechanics consisting of stabiliser states, Clifford unitaries and Pauli measurements. The power of the formalism comes from the description of its elements via simple group theory. Although the origins of the formalism lie in quantum error correction and fault-tolerant quantum computing, the utility of the formalism goes beyond that. The Gottesman-Knill theorem states that the dynamics of stabiliser states under Clifford unitaries and Pauli measurements can be efficiently simulated on a classical computer. This algorithm can be extended to arbitrary states and unitaries in multiple ways at the cost of an increased runtime. This runtime can be seen as a quantification of the non-stabiliser resources needed to implement a quantum circuit. Moreover, as non-stabiliser elements are necessary for universal quantum computing, the runtime provides a way to measure the “non-classicality” of a computation. This is particularly pronounced in the magic state model of quantum computing where the only non-stabiliser elements are given by magic states. Hence, in the resource theory of magic, resources are measured through magic monotones which are operationally linked to runtimes of classical simulation algorithms. In this thesis, I discuss different aspects of the resource theory of magic. The mentioned classical simulation algorithms require the computation of magic monotones which is in general a computationally intractable problem. However, I show that the computational complexity can be exponentially reduced for certain classes of symmetric states, such as copies of magic states. To this end, the symmetries of the convex hull of stabiliser states are characterised and linked to their properties as so-called designs. In addition, I study the recently introduced class of completely stabiliser-preserving channels (CSP), which is the class of quantum channels unable to generate magic resources. It is shown that this class is strictly larger than the class of stabiliser operations, composed of Clifford unitaries and Pauli measurements. This finding could have several interesting consequences. First, it implies that there is a class of efficiently simulable quantum channels beyond the Gottesman-Knill theorem. Second, it is possible that optimal magic state distillation rates cannot be achieved via stabiliser operations and this gap is in fact significant. Further applications of the stabiliser formalism come through design theory. A unitary t-design is an ensemble of unitaries which reproduce the first t moments of the Haar measure on the unitary group. Randomness in the form of Haar-random unitaries is an essential building block in many quantum information protocols. Implementing such Haar-random unitaries is however often impractical. Here, designs can exhibit considerably lower resource requirements while still being random enough for most applications. Some of the most prominent examples of such protocols concern the certification and characterisation of quantum systems, such as randomised benchmarking. Interestingly, the group of Clifford unitaries forms a unitary 3-design and is often the prime choice thanks to efficient group operations. In this thesis, I summarise my recent results obtained with collaborators in constructing approximate unitary t-designs from the Clifford group supplemented by only few non-Clifford gates. Intriguingly, this construction uses only O(t^4) single qubit non-Clifford gates and is independent of the number of qubits n. Overall, this yields a gate count which is a significant improvement over for the Brandao-Harrow- Horodecki construction based on local random circuits. To provide context for this result, I review the representation theory of the Clifford group and define the Clifford semigroup. In an attempt to generalise approximations results for the unitary group to the Clifford group, the Clifford semigroup is investigated for suitable approximations of the Clifford twirl.
Despite the widespread belief that quantum computers cannot be efficiently simulated classically, efficient simulation is known to be possible in certain restricted regimes. In particular, the Gottesman-Knill theorem states that Clifford circuits can be efficiently simulated. We begin this thesis by reviewing and comparing several known techniques for efficient simulation of Clifford circuits: the stabilizer formalism, CH form, affine form, and the graph state formalism. We describe each simulation method and give four different proofs of the Gottesman-Knill theorem. \n \nNext we review a recent work [15], which shows that restricting the geometry of Clifford circuits can lead to a further speedup. We give an algorithm for simulating Pauli basis measurements on a planar graph state in time widetildeO(nomega/2), where omega<2.373 is the matrix multiplication exponent. This algorithm achieves a quadratic speedup over using Clifford simulation methods directly. As an application of this algorithm, we consider a depth-d Clifford circuit whose two-qubit gates act along edges of a planar graph and describe how to sample from its output distribution or compute an output probability in time $\\widetilde{O}(n^{\\omega/2}d^\\omega)$. For d=O(logn), both of these results are quadratic speedups over using Clifford simulation methods directly. \n \nFinally, we extend these simulation algorithms to universal circuits by using stabilizer rank methods. We follow a previously known gadgetization procedure [9] to show that given a depth-d Clifford+T circuit with tT gates and whose two-qubit gates act along edges of a planar graph, we can sample from its output distribution in time widetildeO(20.7926tn5/2t6d3) and can compute output probabilities in time widetildeO(20.3963tn3/2t6d3). Previous work [9,6], applied to the case d=O(logn), gives algorithms for sampling in time O(20.3963tn6t6) and computation of output probabilities in time O(20.3963tn3t3). Our sampling algorithm offers improved scaling in n but poorer scaling in the exponential term, while our algorithm for computing output probabilities offers improved scaling in n with identical scaling in the exponential term.
The classical simulation of quantum dynamics plays an important role in our understanding of quantum complexity and in the development of quantum technologies. Efficient techniques such as those based on the Gottesman-Knill theorem for Clifford circuits, tensor networks for low entanglement-generating circuits, or Wick's theorem for fermionic Gaussian states have become central tools in quantum computing. In this work, we contribute to this body of knowledge by presenting a framework for classical simulations, dubbed “ g -sim ”, which is based on the underlying Lie algebraic structure of the dynamical process. When the dimension of the algebra grows at most polynomially in the system size, there exist observables for which the simulation is efficient. Indeed, we show that g -sim enables new regimes for classical simulations, is able to deal with certain forms of noise in the evolution, as well as can be used to tackle several paradigmatic variational and nonvariational quantum computing tasks. For the former, we perform Lie-algebraic simulations to train and optimize parametrized quantum circuits (thus effectively showing that some variational models can be dequantized), design enhanced parameter initialization strategies, solve tasks of quantum circuit synthesis, and train a quantum-phase classifier. For the latter, we report large-scale noiseless and noisy simulations on benchmark problems. By comparing the limitations of g -sim and certain Wick's theorem-based simulations, we find that the two methods become inefficient for different types of states or observables, hinting at the existence of distinct, nonequivalent resources for classical simulation.
The Gottesman–Knill theorem asserts that quantum circuits composed solely of Clifford gates can be efficiently simulated classically. This theorem hinges on the fact that Clifford gates map Pauli strings to other Pauli strings, thereby allowing for a structured simulation process using classical computations. In this work, we break down the step-by-step procedure of the Gottesman–Knill theorem in a beginner-friendly manner, leveraging concepts such as matrix products, tensor products, commutation, anti-commutation, eigenvalues, and eigenvectors of quantum mechanical operators. Through detailed examples illustrating superposition and entanglement phenomena, we aim to provide a clear understanding of the classical simulation of Clifford gate-based quantum circuits. While we do not provide a formal proof of the theorem, we offer intuitive physical insights at each stage where necessary, empowering readers to grasp the fundamental principles underpinning this intriguing aspect of quantum computation. Quanta 2024; 13: 20–27.