Testing the symmetries of quantum states and channels provides a way to assess their usefulness for different physical, computational, and communication tasks. Here, we establish several complexity-theoretic results that classify the difficulty of symmetry-testing problems involving a unitary representation of a group and a state or a channel that is being tested. In particular, we prove that various such symmetry-testing problems are complete for bounded-error quantum polynomial time, quantum Merlin–Arthur (QMA), quantum statistical zero-knowledge, two-message quantum interactive proofs (QIPs), two-message QIPs restricted to entanglement-breaking provers, and QIPs, thus spanning the prominent classes of the QIP hierarchy and forging a nontrivial connection between symmetry and quantum computational complexity. Finally, we prove the inclusion of two Hamiltonian symmetry-testing problems in QMA and quantum Arthur–Merlin, while leaving it as an intriguing open question to determine whether these problems are complete for these classes.
Testing the symmetries of quantum states and channels provides a way to assess their usefulness for different physical, computational, and communication tasks. Here, we establish several complexity-theoretic results that classify the difficulty of symmetry-testing problems involving a unitary representation of a group and a state or a channel that is being tested. In particular, we prove that various such symmetry-testing problems are complete for bounded-error quantum polynomial time, quantum Merlin–Arthur (QMA), quantum statistical zero-knowledge, two-message quantum interactive proofs (QIPs), two-message QIPs restricted to entanglement-breaking provers, and QIPs, thus spanning the prominent classes of the QIP hierarchy and forging a nontrivial connection between symmetry and quantum computational complexity. Finally, we prove the inclusion of two Hamiltonian symmetry-testing problems in QMA and quantum Arthur–Merlin, while leaving it as an intriguing open question to determine whether these problems are complete for these classes.
Many natural problems in quantum computing involve constructing a quantum state or implementing a unitary transformation. However, relatively little is known about the computational complexity of these problems compared to that of computing boolean functions. In this thesis we do the following:• We prove upper bounds on the complexity of constructing arbitrary states and implementing arbitrary unitaries with the help of a classical oracle. • We prove bounds on the complexity of computing parity in QAC0, a quantum analogue of AC0, by way of a reduction to the task of constructing a certain type of state. • We prove upper bounds on the circuit size and depth required to construct arbitrary states and implement arbitrary unitaries, in various quantum circuit models. Many of these bounds are tight. • We define models of interactive proofs for constructing states and implementing unitaries with the help of an untrusted prover. We prove a quantum state analogue of the inclusion PSPACE ⊆ QIP, and obtain somewhat analogous results for unitaries and with multiple entangled provers. • We present barriers to improving several of the above results.
State transformation problems such as compressing quantum information or breaking quantum commitments are fundamental quantum tasks. However, their computational difficulty cannot easily be characterized using traditional complexity theory, which focuses on tasks with classical inputs and outputs. To study the complexity of such state transformation tasks, we introduce a framework for unitary synthesis problems, including notions of reductions and unitary complexity classes. We use this framework to study the complexity of transforming one entangled state into another via local operations. We formalize this as the Uhlmann Transformation Problem, an algorithmic version of Uhlmann's theorem. Then, we prove structural results relating the complexity of the Uhlmann Transformation Problem, polynomial space quantum computation, and zero knowledge protocols. The Uhlmann Transformation Problem allows us to characterize the complexity of a variety of tasks in quantum information processing, including decoding noisy quantum channels, breaking falsifiable quantum cryptographic assumptions, implementing optimal prover strategies in quantum interactive proofs, and decoding the Hawking radiation of black holes. Our framework for unitary complexity thus provides new avenues for studying the computational complexity of many natural quantum information processing tasks.
Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations. Our QMA proof system is sound against arbitrary quantum provers, but only requires an honest prover to perform polynomial-time quantum computations, provided that it holds a quantum witness for a given instance of the QMA problem under consideration. The proof system relies on a new variant of the QMA-complete local Hamiltonian problem in which the local terms are described by Clifford operations and standard basis measurements. We believe that the QMA-completeness of this problem may have other uses in quantum complexity.
In Chapter 1, we establish the mathematical background used throughout this thesis. We review concepts from group and representation theory. We further establish fundamental concepts from quantum information. This will allow us to then define the different notions of symmetry necessary in the following chapters. In Chapter 2, we investigate Hamiltonian symmetries. We propose quantum algorithms capable of testing whether a Hamiltonian exhibits symmetry with respect to a group. Furthermore, we show that this algorithm is that this algorithm is DQC1-Complete. Finally, we execute one of our symmetry-testing algorithms on existing quantum computers for simple examples. In Chapter 3, we discuss tests of symmetry for quantum states. For the case of testing Bose symmetry of a state, there is a simple and efficient quantum algorithm, while the tests for other kinds of symmetry rely on the aid of a quantum prover. We prove that the acceptance probability of each algorithm is equal to the maximum symmetric fidelity of the state being tested Finally, we establish various generalizations of the resource theory of asymmetry, with the upshot being that the acceptance probabilities of the algorithms are resource monotones. In Chapter 4, we begin by showing that the analytical form of the acceptance probability of such a test is given by the cycle index polynomial of the symmetric group ���� . We derive a family of quantum separability tests, each of which is generated by a finite group; for all such algorithms, we show that the acceptance probability is determined by the cycle index polynomial of the group. Finally, we produce and analyze explicit circuit constructions for these tests, showing that the tests corresponding to the symmetric and cyclic groups can be executed with ��(��2) and ��(�� log(��)) controlled-SWAP gates, respectively, where �� is the number of copies of the state In Chapter 5, we include additional results not previously published; in particular, we give a test for symmetry of a quantum state using density matrix exponentiation, a further result of Hamiltonian symmetry measurements when using Abelian groups, and an alternate Hamiltonian symmetry test construction for a block-encoded Hamiltonian.
Symmetry is a unifying concept in physics. In quantum information and beyond, it is known that quantum states possessing symmetry are not useful for certain information-processing tasks. For example, states that commute with a Hamiltonian realizing a time evolution are not useful for timekeeping during that evolution, and bipartite states that are highly extendible are not strongly entangled and thus not useful for basic tasks like teleportation. Motivated by this perspective, this paper details several quantum algorithms that test the symmetry of quantum states and channels. For the case of testing Bose symmetry of a state, we show that there is a simple and efficient quantum algorithm, while the tests for other kinds of symmetry rely on the aid of a quantum prover. We prove that the acceptance probability of each algorithm is equal to the maximum symmetric fidelity of the state being tested, thus giving a firm operational meaning to these latter resource quantifiers. Special cases of the algorithms test for incoherence or separability of quantum states. We evaluate the performance of these algorithms on choice examples by using the variational approach to quantum algorithms, replacing the quantum prover with a parameterized circuit. We demonstrate this approach for numerous examples using the IBM quantum noiseless and noisy simulators, and we observe that the algorithms perform well in the noiseless case and exhibit noise resilience in the noisy case. We also show that the maximum symmetric fidelities can be calculated by semi-definite programs, which is useful for benchmarking the performance of these algorithms for sufficiently small examples. Finally, we establish various generalizations of the resource theory of asymmetry, with the upshot being that the acceptance probabilities of the algorithms are resource monotones and thus well motivated from the resource-theoretic perspective.
Abstract Problems based on the structure of graphs—for example, finding cliques, independent sets, or colorings—are of fundamental importance in classical complexity. Defining well-formulated decision problems for quantum graphs, which are an operator system generalization of graphs, presents several technical challenges. Consequently, the connections between quantum graphs and complexity have been underexplored. In this work, we introduce and study the clique problem for quantum graphs. Our approach utilizes a well-known connection between quantum graphs and quantum channels. The inputs for our problems are presented as quantum channels induced by circuits, which implicitly determine a corresponding quantum graph. We show that when quantifying over all channels, this problem is complete for QMA(2); in fact, it remains QMA(2) -complete when restricted to channels that are probabilistic mixtures of entanglement-breaking and partial trace channels. Quantifying over a subset of entanglementbreaking channels, this problem becomes QMA-complete, and restricting further to deterministic or classical noisy channels gives rise to complete problems for NP and MA, respectively. In this way, we exhibit a classical complexity problem whose natural quantization is QMA(2), rather than QMA, and provide the first problem that allows for a direct comparison of the classes QMA(2), QMA, MA, and NP by quantifying over increasingly larger families of instances. We use methods that are inspired by self-testing to provide a direct proof of QMA(2)-completeness, rather than reducing to a previously studied complete problem. We also give a new proof of the celebrated reduction of QMA( k ) to QMA(2). In parallel, we study a version of the closely related independent set problem for quantum graphs, and provide preliminary evidence that it may be in general weaker in complexity, contrasting to the classical case.