An official website of the United States government
Abstract Quantum Approximate Optimization algorithm (QAOA) aims to search for approximate solutions to discrete optimization problems with near-term quantum computers. As there are no algorithmic guarantee possible for QAOA to outperform classical computers, without a proof that bounded-error quantum polynomial time (BQP) ≠ nondeterministic polynomial time (NP), it is necessary to investigate the empirical advantages of QAOA. We identify a computational phase transition of QAOA when solving hard problems such as SAT—random instances are most difficult to train at a critical problem density. We connect the transition to the controllability and the complexity of QAOA circuits. Moreover, we find that the critical problem density in general deviates from the SAT-UNSAT phase transition, where the hardest instances for classical algorithms lies. Then, we show that the high problem density region, which limits QAOA’s performance in hard optimization problems (reachability deficits), is actually a good place to utilize QAOA: its approximation ratio has a much slower decay with the problem density, compared to classical approximate algorithms. Indeed, it is exactly in this region that quantum advantages of QAOA over classical approximate algorithms can be identified.
more »
« less
Bosonic Quantum Computational Complexity
In recent years, quantum computing involving physical systems with continuous degrees of freedom, such as the bosonic quantum states of light, has attracted significant interest. However, a well-defined quantum complexity theory for these bosonic computations over infinite-dimensional Hilbert spaces is missing. In this work, we lay the foundations for such a research program. We introduce natural complexity classes and problems based on bosonic generalizations of BQP, the local Hamiltonian problem, and QMA. We uncover several relationships and subtle differences between standard Boolean classical and discrete-variable quantum complexity classes, and identify outstanding open problems. Our main contributions include the following: 1) Bosonic computations. We show that the power of Gaussian computations up to logspace reductions is equivalent to bounded-error quantum logspace (BQL, characterized by the problem of inverting well-conditioned matrices). More generally, we define classes of continuous-variable quantum polynomial time computations with a bounded probability of error (CVBQP) based on gates generated by polynomial bosonic Hamiltonians and particle-number measurements. Due to the infinite-dimensional Hilbert space, it is not a priori clear whether a decidable upper bound can be obtained for these classes. We identify complete problems for these classes, and we demonstrate a BQP lower bound and an EXPSPACE upper bound by proving bounds on the average energy throughout the computation. We further show that the problem of computing expectation values of polynomial bosonic observables at the output of bosonic quantum circuits using Gaussian and cubic phase gates is in PSPACE. 2) Bosonic ground energy problems. We prove that the problem of deciding whether the spectrum of a bosonic Hamiltonian is bounded from below is co-NP-hard. Furthermore, we show that the problem of finding the minimum energy of a bosonic Hamiltonian critically depends on the non-Gaussian stellar rank of the family of energy-constrained states one optimizes over: for zero stellar rank, i.e., optimizing over Gaussian states, it is NP-complete; for polynomially-bounded stellar rank, it is in QMA; for unbounded stellar rank, it is RE-hard, i.e., undecidable.
more »
« less
- Award ID(s):
- 2013062
- PAR ID:
- 10638609
- Editor(s):
- Meka, Raghu
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 325
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 33:1-33:19
- Subject(s) / Keyword(s):
- continuous-variable quantum computing infinite-dimensional quantum systems stellar rank Hamiltonian complexity Theory of computation → Quantum information theory Theory of computation → Quantum complexity theory
- Format(s):
- Medium: X Size: 19 pages; 893120 bytes Other: application/pdf
- Size(s):
- 19 pages 893120 bytes
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Oshman, Rotem (Ed.)In this work, we study the class of problems solvable by (deterministic) Adaptive Massively Parallel Computations in constant rounds from a computational complexity theory perspective. A language L is in the class AMPC⁰ if, for every ε > 0, there is a deterministic AMPC algorithm running in constant rounds with a polynomial number of processors, where the local memory of each machine s = O(N^ε). We prove that the space-bounded complexity class ReachUL is a proper subclass of AMPC⁰. The complexity class ReachUL lies between the well-known space-bounded complexity classes Deterministic Logspace (DLOG) and Nondeterministic Logspace (NLOG). In contrast, we establish that it is unlikely that PSPACE admits AMPC algorithms, even with polynomially many rounds. We also establish that showing PSPACE is a subclass of nonuniform-AMPC with polynomially many rounds leads to a significant separation result in complexity theory, namely PSPACE is a proper subclass of EXP^{Σ₂^{𝖯}}.more » « less
-
Tauman Kalai, Yael (Ed.)We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum"; problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between QMA and QMA(2), a long standing question in quantum complexity theory.more » « less
-
Guruswami, Venkatesan (Ed.)It is a long-standing open question to construct a classical oracle relative to which BQP/qpoly ≠ BQP/poly or QMA ≠ QCMA. In this paper, we construct classically-accessible classical oracles relative to which BQP/qpoly ≠ BQP/poly and QMA ≠ QCMA. Here, classically-accessible classical oracles are oracles that can be accessed only classically even for quantum algorithms. Based on a similar technique, we also show an alternative proof for the separation of QMA and QCMA relative to a distributional quantumly-accessible classical oracle, which was recently shown by Natarajan and Nirkhe.more » « less
-
Guruswami, Venkatesan (Ed.)Relational problems (those with many possible valid outputs) are different from decision problems, but it is easy to forget just how different. This paper initiates the study of FBQP/qpoly, the class of relational problems solvable in quantum polynomial-time with the help of polynomial-sized quantum advice, along with its analogues for deterministic and randomized computation (FP, FBPP) and advice (/poly, /rpoly). Our first result is that FBQP/qpoly ≠ FBQP/poly, unconditionally, with no oracle - a striking contrast with what we know about the analogous decision classes. The proof repurposes the separation between quantum and classical one-way communication complexities due to Bar-Yossef, Jayram, and Kerenidis. We discuss how this separation raises the prospect of near-term experiments to demonstrate "quantum information supremacy," a form of quantum supremacy that would not depend on unproved complexity assumptions. Our second result is that FBPP ̸ ⊂ FP/poly - that is, Adleman’s Theorem fails for relational problems - unless PSPACE ⊂ NP/poly. Our proof uses IP = PSPACE and time-bounded Kolmogorov complexity. On the other hand, we show that proving FBPP ̸ ⊂ FP/poly will be hard, as it implies a superpolynomial circuit lower bound for PromiseBPEXP. We prove the following further results: - Unconditionally, FP ≠ FBPP and FP/poly ≠ FBPP/poly (even when these classes are carefully defined). - FBPP/poly = FBPP/rpoly (and likewise for FBQP). For sampling problems, by contrast, SampBPP/poly ≠ SampBPP/rpoly (and likewise for SampBQP).more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/lipics.itcs.2025.33
