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Exact requirements for battery-assisted qubit gates

Riccardo Castellano, Vasco Cavina, Martí Perarnau-Llobet, Pavel Sekatski, Vittorio Giovannetti

Abstract

We consider the implementation of a unitary gate on a qubit system S via a global energy-preserving operation acting on S and an auxiliary system B that can be seen as a battery. We derive a simple, asymptotically exact expression for the implementation error as a function of the battery state, which we refer to as the it Unitary Defect. Remarkably, this quantity is independent of the specific gate being implemented, highlighting a universal property of the battery itself. We show that minimizing the unitary defect, under given physical constraints on the battery state, is mathematically equivalent to solving a Lagrangian optimization problem, often corresponding to finding the ground state of a one-dimensional quantum system. Using this mapping, we identify optimal battery states that achieve the highest precision under constraints on energy, squared energy, number of levels and Quantum Fisher Information. Overall, our results provide an efficient method for establishing bounds on the physical requirements needed to implement a unitary gate via energy-preserving operations and for determining the corresponding optimal protocols.

Exact requirements for battery-assisted qubit gates

Abstract

We consider the implementation of a unitary gate on a qubit system S via a global energy-preserving operation acting on S and an auxiliary system B that can be seen as a battery. We derive a simple, asymptotically exact expression for the implementation error as a function of the battery state, which we refer to as the it Unitary Defect. Remarkably, this quantity is independent of the specific gate being implemented, highlighting a universal property of the battery itself. We show that minimizing the unitary defect, under given physical constraints on the battery state, is mathematically equivalent to solving a Lagrangian optimization problem, often corresponding to finding the ground state of a one-dimensional quantum system. Using this mapping, we identify optimal battery states that achieve the highest precision under constraints on energy, squared energy, number of levels and Quantum Fisher Information. Overall, our results provide an efficient method for establishing bounds on the physical requirements needed to implement a unitary gate via energy-preserving operations and for determining the corresponding optimal protocols.

Paper Structure

This paper contains 14 sections, 5 theorems, 144 equations, 1 figure, 1 table.

Table of Contents

  1. Choi infidelity
  2. Equivalence of worst-case infidelity and Choi infidelity
  3. Calculation of the Choi infidelity
  4. Comments on the assumptions on the battery input state and the global interaction
  5. Analysis of the ground state of the battery
  6. On the optimality of the unitary interaction between S and B
  7. Minimizing the Choi infidelity using $N$ levels via discrete sine transform
  8. Explicit optimization of the precision at fixed battery resources
  9. Minimizing the Choi infidelity at fixed $\langle E^{2}\rangle$
  10. Minimizing the Choi infidelity at fixed mean energy
  11. Minimizing the Choi infidelity at fixed QFI
  12. Performances of coherent states
  13. Qudit target systems
  14. An alterative scheme for gates with invariant subspaces block diagonal in energy.

Key Result

Proposition 1

Let $\mathbf{\Phi}$ be an arbitrary CPTP channel defined on an Hilbert space of dimension $d$ and $\mathcal{V} := V\cdots V^{\dag}$ a unitary mapping on the same space, then

Figures (1)

  • Figure 1: Schematic representation of the approximated implementation of a non-energy-preserving gate (NEPG) $V_S$ on the system $S$ through a joint total-energy-preserving unitary transformation $U_{SB}$ with the battery $B$ initialized in the input state $\beta_B$.

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Corollary 4
  • proof
  • Proposition 5
  • proof