Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symmetry and Topology of Successive Quantum Feedback Control

Junxuan Wen, Zongping Gong, Takahiro Sagawa

Abstract

We establish a symmetry classification for a general class of quantum feedback control. For successive feedback control with a non-adaptive sequence of bare measurements (i.e., with positive Kraus operators), we prove that the symmetry classification collapses to the ten-fold AZ$^\dagger$ classes, specifying the allowed topology of CPTP maps associated with feedback control. We demonstrate that a chiral Maxwell's demon with Gaussian measurement errors exhibits quantized winding numbers. Moreover, for general (non-bare) measurements, we explicitly construct a protocol that falls outside the ten-fold classification. These results broaden and clarify the principles in engineering topological aspects of quantum control robust against disorder and imperfections.

Symmetry and Topology of Successive Quantum Feedback Control

Abstract

We establish a symmetry classification for a general class of quantum feedback control. For successive feedback control with a non-adaptive sequence of bare measurements (i.e., with positive Kraus operators), we prove that the symmetry classification collapses to the ten-fold AZ classes, specifying the allowed topology of CPTP maps associated with feedback control. We demonstrate that a chiral Maxwell's demon with Gaussian measurement errors exhibits quantized winding numbers. Moreover, for general (non-bare) measurements, we explicitly construct a protocol that falls outside the ten-fold classification. These results broaden and clarify the principles in engineering topological aspects of quantum control robust against disorder and imperfections.

Paper Structure

This paper contains 2 sections, 3 theorems, 23 equations, 4 figures, 1 table.

Table of Contents

  1. Insensitivity of PBC spectrum to the system size
  2. Behavior against Gaussian tail cutoff

Key Result

Theorem 1

Consider a family of CPTP maps of feedback control $\left\{ \mathcal{E}_{\tau_1,\cdots,\tau_N} \right\}$ in the form of Eqs.(general form of feedback control)-(unitary with feedback Hamiltonian). If every measurement in the protocol is bare, then the possible BL symmetry classes of $\mathcal{E}_{\ta

Figures (4)

  • Figure 1: Schematic of successive feedback control with non-adaptive measurements. The Kraus operators $\{ \hat{M}_m^{(i)} \}$ of the $i$-th measurement can depend on $i$ but are fixed in advance, and the feedback is applied via feedback Hamiltonians conditioned on previous outcomes.
  • Figure 2: Spectra of the CPTP maps of chiral Maxwell's demon with projective measurement (a) and that with Gaussian errors (b.1) -- (b.4). The colored dots are the eigenvalues under PBC for size $L=400$, whose colors correspond to the wavenumber $k$ of the Bloch matrices, and the black dots are the eigenvalues under OBC for size $L = 20$. The unit of energy, the height of the potential and the feedback duration are set to be $J=1,V=10^2$, $\tau=1$ respectively. We set the radius of the error (cutoff) to be $c=2$, and the Gaussian sharpness is set to be $\ell=15$. The winding number $w$ around the origin is $w(\xi_{\text{PG}}=0)=-1$ for the left and $w(\xi_{\text{PG}}=0)=0$ for the right, and that around $1-\epsilon$ is $w(\xi_{\text{PG}}=1-\epsilon)=-1$ for both pictures with $\epsilon = 10^{-12}$.
  • Figure 3: The PBC spectra of the CPTP maps of chiral Maxwell's demon with Gaussian errors for system size $L=50, 100, 200$. The Gaussian sharpness is taken to be $\ell=15$$(30)$ for the left (right). The color of PBC spectrum corresponds to the wavenumber $k$ of the Bloch matrices. The unit of energy, the height of the potential and the feedback duration are set to be $J=1, V=10^4$, $\tau=1$, respectively. We set the radius of the error (cutoff) to $c=2$.
  • Figure 4: The dependence of PBC spectra against the Gaussian tail cutoff $c$ of the measurement error in chiral Maxwell's demon. The colored dots are the eigenvalues under PBC for $L=400$, whose colors correspond to the wavenumber $k$ of the Bloch matrices. The unit of energy, the height of the potential and the feedback duration are set to be $J=1, V=10^4$, $\tau=1$, respectively. The Gaussian sharpness is taken to be $\ell=15$.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • Lemma A
  • proof