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Uniqueness of imaginarity-assisted transformation from computationally universal to strictly universal quantum computation

Yasuaki Nakayama, Yuki Takeuchi, Seiseki Akibue

Abstract

The computational universality with an elementary gate set $\{H,CCZ\}$ can be transformed to the strict universality by using a maximally imaginary state $|+i\rangle$ and some non-imaginary ancillary qubits. From the viewpoint of operational resource theory, it would be intriguing to elucidate a resource for the universality transformation. In this paper, we explore a necessary and sufficient condition for resource states to realize the universality transformation under free real operations. We show that $|+i\rangle$ is a unique resource state up to the free operations. Moreover, we obtain a stronger conclusion. If a given resource state cannot be used for the universality transformation, then realizable quantum gates are restricted to real orthogonal matrices. Therefore, we can tell that $|+i\rangle$ is unique (up to the free operations) not only as a state whose resource measure of imaginarity is maximal, but also as a state which empowers real operations with the ability to apply at least one non-real quantum gate (regardless of the magnitudes of its imaginary parts).

Uniqueness of imaginarity-assisted transformation from computationally universal to strictly universal quantum computation

Abstract

The computational universality with an elementary gate set can be transformed to the strict universality by using a maximally imaginary state and some non-imaginary ancillary qubits. From the viewpoint of operational resource theory, it would be intriguing to elucidate a resource for the universality transformation. In this paper, we explore a necessary and sufficient condition for resource states to realize the universality transformation under free real operations. We show that is a unique resource state up to the free operations. Moreover, we obtain a stronger conclusion. If a given resource state cannot be used for the universality transformation, then realizable quantum gates are restricted to real orthogonal matrices. Therefore, we can tell that is unique (up to the free operations) not only as a state whose resource measure of imaginarity is maximal, but also as a state which empowers real operations with the ability to apply at least one non-real quantum gate (regardless of the magnitudes of its imaginary parts).
Paper Structure (4 sections, 2 theorems, 20 equations, 1 figure)

This paper contains 4 sections, 2 theorems, 20 equations, 1 figure.

Table of Contents

  1. Appendix A: Proof of the $\ket{\psi}$-independence of $\rho^\prime$
  2. Appendix B: Proof of the equivalence of $\tr[\rho \rho^\ast] = 0$ and $||\rho - \rho^\ast||_1 = 2$
  3. Appendix C: Proof of the equivalence between universal resource and maximally imaginary states, which can be transformed to $\ket{\hat{+}}$ by a real CPTP map
  4. Appendix D: Remark on the construction of a real operation from $\rho$ to $\ket{\hat{+}}$

Key Result

Theorem 1

$\rho$ is not zero resource if and only if $\rho$ is universal resource.

Figures (1)

  • Figure 1: Naively, there is expected to be an intermediate region between universal resource and zero resource, but actually there is no such region and there are only two types: universal and zero resources. $\mathbf{Left}:$ Naive expectation. $\mathbf{Right}:$ Actual situation.

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Theorem 1: Main theorem of this paper
  • proof
  • Proposition 1
  • proof
  • proof
  • proof
  • proof