Bosonic Quantum Computational Complexity

TL;DR

The paper lays a foundational complexity theory for bosonic continuous-variable quantum computations, defining CV analogs of BQP and QMA and analyzing their relations to discrete-variable classes. It distinguishes Gaussian (d ≤ 2) and non Gaussian (d > 2) dynamics, establishing a tight link between Gaussian CV dynamics and bounded-error quantum logspace (GDC = BQL) and showing that non Gaussian CV computations can realize BQP and, under various energy and rank constraints, belong to EXPSPACE or even undecidable regimes. It introduces the continuous-variable local Hamiltonian problem (CVLH) and its dependence on stellar rank, proving NP-hardness, QMA-hardness, or undecidability as the non Gaussian character and resource limits vary. An extensive set of results ties energy growth, gate degree, and representation theory (stellar rank) to computational power, and provides parallel algorithms to compute certain CV observables within PSPACE. The work also opens avenues for CVQMA and continuous-variable history-state constructions, highlighting deep connections to the extended Church–Turing thesis and real-number complexity, with implications for both theory and physically realizable CV quantum systems.

Abstract

Quantum computing involving physical systems with continuous degrees of freedom, such as the quantum states of light, has recently 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 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. In particular: 1. We show that the power of quadratic (Gaussian) quantum dynamics is equivalent to the class BQL. More generally, we define classes of continuous-variable quantum polynomial time computations with a bounded probability of error based on higher-degree gates. 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 demonstrate a BQP lower and EXPSPACE upper bound. We further show that the problem of computing expectation values of polynomial bosonic observables is in PSPACE. 2. We prove that the problem of deciding the boundedness of the spectrum of a bosonic Hamiltonian 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 constant stellar rank, it is NP-complete; for polynomially-bounded rank, it is in QMA; for unbounded rank, it is undecidable.

Figures (6)

• Figure 1: Known relations between complexity classes used in this work. If a line connects complexity class $\mathbf C_1$ to $\mathbf C_2$, with $\mathbf C_1$ being on the right of $\mathbf C_2$, it is implied that $\textbf{C}_2 \subseteq \textbf{C}_1$.
• Figure 2: Relations between complexity classes and computational problems proven in this paper and described below, up to logspace reductions on the left and polynomial-time reductions on the right. If a line connects two complexity classes, it is implied that the one below in the diagram is included in the one above. If a line connects a problem to a complexity class, it is implied that the problem is in that class if it is below, and hard for that class if it above. Square brackets indicate specific choices of non-Gaussian gates, where $X^3$ and $\bar{X}$ are polynomial Hamiltonians of degree $3$, while $\bar{Z}\otimes\bar{Z}$ is a polynomial Hamiltonian of degree $4$. The other inclusions are given by the corresponding theorems or trivial from the definitions of the classes.
• Figure 3: Energy bound conditions for quantum computations. The red curve depicts a system that is promised to have low energy at the beginning and the end of the computation, but may have arbitrary high energy in between. The blue curve depicts a model where we impose energy restrictions at any point during the computation.
• Figure 4: A schematic of an instance of evolution for the problem of Gaussian evolution described in \ref{['def:Gaussian-simulation']}. We allow multiple Hamiltonians, and measurements in the end, starting from the vacuum state $\ket{0^m}$. Note that we require the energy of the computation to be bounded at all times by $E^\ast = O(\mathsf{poly}(m))$, and that the total evolution time is bounded by a polynomial, while Hamiltonian coefficients are held constant.
• Figure 5: An example of the circuit for computing sample mean of many Gaussian circuits. Here we have shown an example where there are $4$ Gaussian circuits $\mathcal{C}^{(1)}, \cdots \mathcal{C}^{(4)}$, however, as explained in the proof of \ref{['prop:sample-mean']} this approach can be extended to many circuits. We highlight that in case of having $2^r$ initial circuits, we require $2^{r}-1$ many beam splitters. Note that the circuit composes of beam splitters only, and therefore, leaves the energy invariant. Moreover, the 50:50 beam splitter is defined as in \ref{['eq:beam-splitter']}, where the transformation in the Heisenberg picture follows $X_1\mapsto \frac{1}{\sqrt 2}(X_1 + X_2)$ and $X_2 \mapsto \frac{1}{\sqrt 2}(-X_1 + X_2)$. Therefore, we note that the average (up to scaling) is computed on the top mode.

Theorems & Definitions (151)

• Definition 1.1: Gaussian dynamical computations, informal
• Theorem 1.1: The computational power of Gaussian dynamics, informal
• Definition 1.2: $\mathbf{CVBQP}{[}X^3{]}$, informal
• Theorem 1.2: Upper bound on the computational power of Gaussian and cubic phase gates, informal
• Theorem 1.3: The computational complexity of bosonic expectation values, informal
• proof : Proof sketch
• Theorem 1.4: Complexity of the boundedness problem, informal
• proof : Proof sketch
• Proposition 1.1: Checking boundedness for degree-4 Hamiltonians, informal
• Definition 1.3: The continuous-variable local Hamiltonian problem, informal