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Hamiltonian quantum gates -- energetic advantage from entangleability

Josey Stevens, Sebastian Deffner

TL;DR

This work quantifies the fundamental energy cost of implementing Hamiltonian quantum gates under classical field control by deriving a lower bound on the control-field energy tied to gate error. Using a tensor-product decomposition to represent multipartite gates, the authors relate field fluctuations to a set of observable coefficients $\lambda_i$, whose variances $\epsilon_i$ bound gate infidelity, with $\epsilon \le \sum_i |\epsilon_i|$. They show the energy required scales as $\langle E \rangle \ge \frac{\hbar \omega_0}{4 \epsilon^2} \max_i |\lambda_i|^2$ (or additively for independent terms), implying higher precision demands more energy. Crucially, they demonstrate that entangleability of the generating Hamiltonian can dramatically reduce energy costs, and with suitably large ancilla resources, universal quantum computation could, in principle, be performed with vanishing external energy—transferring control load to system interactions. The results emphasize entangleability as a control resource and illuminate fundamental energy-tradeoffs in quantum computing, while acknowledging practical limitations such as dissipation and exact multi-qubit interactions.

Abstract

Hamiltonian quantum gates controlled by classical electromagnetic fields form the basis of any realistic model of quantum computers. In this letter, we derive a lower bound on the field energy required to implement such gates and relate this energy to the expected gate error. We study the entangleability (ability to entangle qubits) of Hamiltonians and highlight how this feature of quantum gates can provide a means for more energetically efficient computation. Ultimately, we show that a universal quantum computer can be realized with vanishingly low energetic requirements but at the expense of arbitrarily large complexity.

Hamiltonian quantum gates -- energetic advantage from entangleability

TL;DR

This work quantifies the fundamental energy cost of implementing Hamiltonian quantum gates under classical field control by deriving a lower bound on the control-field energy tied to gate error. Using a tensor-product decomposition to represent multipartite gates, the authors relate field fluctuations to a set of observable coefficients , whose variances bound gate infidelity, with . They show the energy required scales as (or additively for independent terms), implying higher precision demands more energy. Crucially, they demonstrate that entangleability of the generating Hamiltonian can dramatically reduce energy costs, and with suitably large ancilla resources, universal quantum computation could, in principle, be performed with vanishing external energy—transferring control load to system interactions. The results emphasize entangleability as a control resource and illuminate fundamental energy-tradeoffs in quantum computing, while acknowledging practical limitations such as dissipation and exact multi-qubit interactions.

Abstract

Hamiltonian quantum gates controlled by classical electromagnetic fields form the basis of any realistic model of quantum computers. In this letter, we derive a lower bound on the field energy required to implement such gates and relate this energy to the expected gate error. We study the entangleability (ability to entangle qubits) of Hamiltonians and highlight how this feature of quantum gates can provide a means for more energetically efficient computation. Ultimately, we show that a universal quantum computer can be realized with vanishingly low energetic requirements but at the expense of arbitrarily large complexity.

Paper Structure

This paper contains 10 sections, 27 equations, 1 figure.

Table of Contents

  1. Coherent Hamiltonian quantum control
  2. Tensor product decomposition
  3. Sub-linearity of gate errors
  4. Required energy
  5. Example: energetic cost of single qubit gates
  6. Energetic benefits of entangleability
  7. Universal quantum computation
  8. Concluding remarks
  9. Deriving the control error
  10. Upper bound on the achieved gate error

Figures (1)

  • Figure 1: The Renyi-$1/2$ entropy over the duration of the driving protocol implemented by Hamiltonians \ref{['eq Hxx1']} (blue) and \ref{['eq Hxx2']} (orange). The initial state is prepared such that both qubits are in the zero state, i.e., $\rho = \ket{0 0}\bra{0 0}$ and are driven by electromagnetic fields consisting of a single mode with $\omega_0 = \frac{1}{\tau}$. On the inset we show the path on the Bloch sphere of the first qubit taken at equal timesteps of $\tau/10$.