Random unitaries that conserve energy

TL;DR

The paper studies energy-conserving pseudorandom unitaries (PRUs) as a refinement of Haar-random unitaries under a fixed Hamiltonian $H$. It proves a dichotomy: energy-conserving PRUs exist for random commuting Hamiltonians via a QPE-based construction with pseudorandom phases, but are provably non-existent for certain 1D translationally invariant Hamiltonians, where an efficient quantum distinguisher separates energy-conserving Haar unitaries from any poly-size circuit under cryptographic assumptions. It further shows that deciding existence of energy-conserving PRUs for a given Hamiltonian family is undecidable, by embedding the halting problem into the Hamiltonian construction. The results connect quantum complexity, cryptography, and Hamiltonian dynamics, illustrating a fundamental computational barrier between generic random unitaries and physically constrained, energy-conserving dynamics. These insights have implications for modeling chaotic quantum dynamics under energy conservation and for understanding the limits of simulating energy-constrained quantum systems.

Abstract

Random unitaries sampled from the Haar measure serve as fundamental models for generic quantum many-body dynamics. Under standard cryptographic assumptions, recent works have constructed polynomial-size quantum circuits that are computationally indistinguishable from Haar-random unitaries, establishing the concept of pseudorandom unitaries (PRUs). While PRUs have found broad implications in many-body physics, they fail to capture the energy conservation that governs physical systems. In this work, we investigate the computational complexity of generating PRUs that conserve energy under a fixed and known Hamiltonian $H$. We provide an efficient construction of energy-conserving PRUs when $H$ is local and commuting with random coefficients. Conversely, we prove that for certain translationally invariant one-dimensional $H$, there exists an efficient quantum algorithm that can distinguish truly random energy-conserving unitaries from any polynomial-size quantum circuit. This establishes that energy-conserving PRUs cannot exist for these Hamiltonians. Furthermore, we prove that determining whether energy-conserving PRUs exist for a given family of one-dimensional local Hamiltonians is an undecidable problem. Our results reveal an unexpected computational barrier that fundamentally separates the generation of generic random unitaries from those obeying the basic physical constraint of energy conservation.

Figures (7)

• Figure 1: (a) Haar random unitary and energy-conserving Haar random unitary. Haar random unitaries are those that scramble the full Hilbert space. They transit each wavefunction to the infinite-temperature state. In contrast, energy-conserving Haar random unitaries are those that only scramble degenerated subspaces. They respect the energy-occupation of each wavefunction. (b) Summary of our results. We construct local Hamiltonian families with and without energy-conserving PRU. Further, we demonstrate there exists a certian set of Hamiltonian families such that determining if they have energy-conserving PRU is an undecidable problem. (c) We construct a universal distinguishing algorithm to prove Theorem \ref{['main:thm:hard-ham']}. The algorithm accepts the energy-conserving Haar random unitaries of the hard Hamiltonian, whereas rejects any polynomial-size quantum circuit.
• Figure 2: (a) Illustration of Turing machines. A TM consists of an infinite tape storing symbols, a read/write head that moves left or right who has a internal state, and a set of transition rules to determine the actions and movements of the head. (b) To represent any configuration of TMs (with finite memory size) with a product state, we first put the head state into the tape to make the whole system strictly one dimensional. Then the configuration can be represented by a quantum product state with local Hilbert spaces containing all the symbols and internal states. (c) The idea of using energy-conserving Haar-random unitaries to do fast computation: any computation process can be viewed as moving along a one-dimensional path with vertices labeled by TM's configurations. After initialize the wavefunction to the left terminal corresponding to inputs, the Haar-random unitaries scramble the wavefunction to disperse along the whole path. Then a subsequent measurement collapses the wavefunction to any vertex with almost equal probability, making an exponentially large step forward at once. This idealized picture faces practical obstacles, see Section \ref{['main:hard-ham']}.
• Figure 3: Turing machine that generates a Hamiltonian family $H_n^{(x)}$ with or without energy-conserving PRU, depending on whether universal TM halts upon input $x$. After receiving an put $n$, two machines run in parallel: $\mathsf{M}_1$ generates the hard Hamiltonian $H$ from Section \ref{['main:hard-ham']}, and $\mathsf{M}_2$ is a UTM operating upon input $x$. If $\mathsf{M}_1$ halts, then the whole system halts with output $H$. If $\mathsf{M}_2$ halts, $\mathsf{M}_1$ halts and does the reverse computing. Thus the system outputs $0$.
• Figure 4: Circuit structure for quantum phase estimation.
• Figure 5: Entire construction of energy-conserving PRU for random commuting Hamiltonians.

Theorems & Definitions (83)

• Definition 1: Pseudorandom unitaries; informal
• Definition 2: Energy-Conserving PRUs; informal
• Theorem 1: Constructing energy-conserving PRUs for random commuting Hamiltonians
• Theorem 2: Hard Hamiltonians with no energy-conserving PRUs
• Theorem 3: Undecidability of existence of energy-conserving PRUs
• Lemma 1: Energy-conserving random unitary can be used to solve $\mathsf{PSPACE}$ problems, Theorem \ref{['thm:ru-solve-pspace']} in Appendix \ref{['ap:hardness-ham-construction']}
• Lemma 2: Distinguishing the TQBF solver, Theorem \ref{['thm:tqbf-verification']} in Appendix \ref{['ap:verify-tqbf-via-oneway']}
• Definition 1: Haar ensemble
• Definition 2: Pseudorandom unitaries
• Definition 3: Energy-conserving pseudorandom unitary ensemble