This project focused on upper and lower bounds for quantum computability using constant depth quantum circuits, and on the complexity theory of such circuits. It established significant differences between resource bounded quantum and classical computation models, particularly emphasizing new examples of where quantum circuits are more powerful than their classical counterparts. A second focus and outgrowth of this work was the creation of efficient quantum algorithms for specific problems and central combinatorial functions. Recent finding include upper bounds for the computational power of constant depth circuits composed of single qubit and CNOT gates, and lower bounds for the computational power of constant depth circuits with single qubit, CNOT and fanout gates. These results represent the power and the limits of small, possibly realizable quantum circuits. Also examined were bounds on computations with additional storage qubits (ancillae), and natural simple functions (realized by small depth circuits) which require ancillae for their computation.
This project focused on upper and lower bounds for quantum computability using constant depth quantum circuits, and on the complexity theory of such circuits. It established significant differences between resource bounded quantum and classical computation models, particularly emphasizing new examples of where quantum circuits are more powerful than their classical counterparts. A second focus and outgrowth of this work was the creation of efficient quantum algorithms for specific problems and central combinatorial functions. Recent finding include upper bounds for the computational power of constant depth circuits composed of single qubit and CNOT gates, and lower bounds for the computational power of constant depth circuits with single qubit, CNOT and fanout gates. These results represent the power and the limits of small, possibly realizable quantum circuits. Also examined were bounds on computations with additional storage qubits (ancillae), and natural simple functions (realized by small depth circuits) which require ancillae for their computation.
We have seen in the previous chapters that quantum computers seem to be more powerful than classical computers for certain problems. There are limits on the power of quantum computers, however. Since a classical computer can simulate a quantum computer, a quantum computer can only compute the same set of functions that a classical computer can. The advantage of using a quantum computer is that the amount of resources needed by a quantum algorithm might be much less than what is needed by the best classical algorithm. In Section 9.1 we briefly define some classical and quantum complexity classes and give some relationships between them. Most of the interesting questions relating classical and quantum complexity classes remain open. For example, we do not yet know if a quantum computer is capable of efficiently solving an NP-complete problem (defined later). One can prove upper bounds on the difficulty of a problem by providing an algorithm that solves that problem, and proving that it will work within in a given running time. But how does one prove a lower bound on the computational complexity of a problem? For example, if we wish to find the product of two n-bit numbers, computing the answer requires outputting roughly 2n bits and that requires Ω(n) steps (in any computing model with finite-sized gates). The best-known upper bound for integer multiplication is O(n log n log log n) steps. It has proved extremely difficult to derive non-trivial lower bounds on the computational complexity of a problem. Most of the known non-trivial lower bounds are in the ‘black-box’ model (for both classical and quantum computing), where we only query the input via a ‘black-box’ of a specific form. We discuss the black-box model in more detail in Section 9.2. We then sketch several approaches for proving black-box lower bounds. The first technique has been called the ‘hybrid method’ and was used to prove that quantum searching requires Ω(√n) queries to succeed with constant probability. The second technique is called the ‘polynomial method’. We then describe a technique based on ‘block sensitivity’, and conclude with a technique known as the ‘adversary method’.
Quantum computation was proposed as an alternate computing process in the late-20th century, when classical computing models faced limitation in solving quantum mechanical problems. Major breakthroughs came in the 1990s, when quantum algorithms for integer factoring and black-box searching were invented that are faster than the best classical algorithms known even today. In this thesis, we look at quantum circuits with limited depth and size and analyse their power and limitations.
Computer Science is about the study of computational problems, many of which are motivated by real-world applications. We are interested as much in an actual method to solve a problem as in its unsolvability. In computational complexity theory we separate out the actual problem from the computing apparatus (e.g. a Turing machine, an AND-OR circuit) and analyse the hardness of the computation, due both to the intrinsic complexity of the problem and to the underlying model.
In this thesis, we consider quantum circuits built with a universal set of gates and show that non-constant depth is required to compute a particular function (parity) and other similar functions. The parity function, extensively studied previously, was shown to take non-constant depth for classical circuits too, thus providing an evidence of an inherent hardness in the function itself. On the other hand we show that the quantum circuit model is rich enough to build a universal circuit (one circuit that can simulate all circuits) that has roughly the same depth or size as the simulated circuit. We also consider a fault model in which some of the inputs to a gate might not be connected to the gate's output. We ask if we can detect the faulty inputs by testing the gate on different inputs. Our goal is to minimise the number of tests. We show that this is very hard for general gates but for certain gates it can be provably easier to detect faulty inputs. We even show that quantum algorithms can provably outperform classical algorithms for these gates. These results give us a sense of wherein lies the power of quantum computation and quantum circuits in particular.
In this work, we prove the strongest known lower bounds for QAC0, allowing polynomially many gates and ancillae. Our main results show that:
(1) Depth-3 QAC0 circuits cannot compute PARITY, and require Ω(exp(n)) gates to compute MAJORITY.
(2) Depth-2 circuits cannot approximate high-influence Boolean functions (e.g., PARITY) with non-negligible advantage, regardless of size.
We develop new classical simulation techniques for QAC0 to obtain our depth-3 bounds. In these results, we relax the output requirement of the quantum circuit to a single bit, making our depth 2 approximation bound stronger than the previous best bound of Rosenthal (2021). This also enables us to draw natural comparisons with classical AC0 circuits, which can compute PARITY exactly in depth 2 (exp size). Our techniques further suggest that, for boolean total functions, constant-depth quantum circuits do not necessarily provide more power than their classical counterparts. Our third result shows that depth 2 QAC0 circuits, regardless of size, cannot exactly synthesize an n-target nekomata state (a state whose synthesis is directly related to the computation of PARITY). This complements the depth 2 exponential size upper bound of Rosenthal (2021) for approximating nekomatas (which is used as a sub-circuit in the only known constant depth PARITY upper bound). Finally, we argue that approximating PARITY in QAC0, with significantly better than 1/poly(n) advantage on average, is just as hard as computing it exactly. Thus, extending our techniques to higher depths would also rule out approximate circuits for PARITY and related problems
Quantum computers were initially proposed to surpass the limitations of classical computers in simulating physical systems. Soon after, the field of quantum computing advanced through landmark discoveries of quantum algorithms, with some conjectured to provide exponential advantages over classical counterparts. These include Shor’s factoring algorithm, which has profound implications in cryptography. Additionally, moderate yet significant polynomial separations have further increased interest in the field. However, despite this progress, unconditional proofs confirming quantum superiority on the theoretical side remain elusive. At the same time, large, noise-resilient quantum computations on the experimental side, which could challenge the extended Church-Turing thesis by potentially breaking classical computational barriers, have yet to be achieved. Meanwhile, our understanding of the differences between classical and quantum mechanics has deepened. We now possess rigorous theoretical tools, such as Bell inequalities and nonlocal games, to describe behaviors unique to these theories. Moreover, the main loopholes for observing non-classical correlations have been closed experimentally, showing that correlations in the physical world exceed those predicted by classical mechanics. This thesis investigates computational complexity differences between quantum and classical models by studying shallow-depth circuits. The primary focus lies in identifying, elucidating, and proving computational quantum advantages. Additionally, the analysis delves into the bijection between these circuits and foundational non-local game constructions, exploring how quantum-classical deviations in multi-party correlation settings translate to computational advantages. A key question addressed is how to synthesize constant-depth quantum circuits for various problems to leverage quantum advantages. To this end, circuit synthesis techniques tailored to measurement-based quantum computation are developed, enabling simultaneously optimal quantum strategies for certain nonlocal games based on a particular bijection between this model and XOR non-local games. Building on this work, we confront a pivotal challenge: If a candidate quantum circuit potentially provides a computational advantage, how can we prove that no classical circuit surpasses it? This work presents new unconditional quantum-classical separations within shallow-depth settings, distinguishing for the first time constant-depth quantum circuits with one- and two-qubit gates from biased polynomial threshold circuits, which incorporate neural-network-type functionalities. These separations extend to prime-dimensional quantum systems, and we conjecture to have identified Razborov-Smolensky-type problems uniquely solvable by specific prime dimensions. The thesis also differentiates constant-depth quantum circuits with multi-qubit Toffoli gates from their classical counterparts, emphasizing the power of parallel quantum computation. Finally, verifying shallow-depth quantum computations is challenging due to conjectured exponential separations from classical computations, limiting correctness verification through simulation. While specific quantum algorithms within BQP have been addressed, we explore whether a quantum computer can reliably execute any instance from a fixed class of quantum circuits. Moving beyond heuristic-based solutions, we propose a benchmarking protocol that certifies, with exponentially high confidence, the execution of arbitrary circuits of specified dimensions, widths, and depths.
We consider the resource bounded quantum circuit model with circuits restricted by the number of qubits they act upon and by their depth. Focusing on natural universal sets of gates which are familiar from classical circuit theory, several new lower bounds for constant depth quantum circuits are proved. The main result is that parity (and hence fanout) requires log depth quantum circuits, when the circuits are composed of single qubit and arbitrary size Toffoli gates, and when they use only constantly many ancill\ae. Under this constraint, this bound is close to optimal. In the case of a non-constant number a of ancill\ae\ and n input qubits, we give a tradeoff between a and the required depth, that results in a non-constant lower bound for fanout when a=n1−o(1). We also show that, regardless of the number of ancill\ae\, arbitrary arity Toffoli gates cannot be simulated exactly by a constant depth circuit family with gates of bounded arity.