Monitoring of quantum walks with weak measurements

1. Problem Statement

The paper addresses the statistical properties of quantum walks under continuous or repeated monitoring. While it is well-established that strong (projective) measurements lead to quantized mean return times governed by a topological winding number ($n_w$), the behavior of these systems under weak measurements remains less understood.

Key questions include:

• Does the quantization of the mean return time survive when projective measurements are replaced by weak, indirect measurements?
• How does the measurement strength ($\eta$) affect the return time statistics and the connection to unitary evolution?
• Can a systematic theoretical framework be developed to link weak monitoring to the underlying topology of the quantum state?

2. Methodology

The authors employ a framework of indirect measurements using an ancilla system, modeled via Kraus operators and generalized measurements.

• Model Setup:

  • A quantum walk evolves via a unitary operator $U_\tau = e^{-iH\tau}$ over a fixed time step $\tau$.
  • After each step, an indirect measurement is performed with tunable strength $\eta$ ($0 < \eta \le 1$).
  • The measurement is implemented by coupling the system to an ancilla (via controlled-$R_Y$ gates), resulting in a "sheared" projection operator.
  • The projection operators are defined as:
    $$\begin{pmatrix} P_\eta \\ Q_\eta \end{pmatrix} = \begin{pmatrix} \sqrt{\eta} & 0 \\ \sqrt{1-\eta} & 1 \end{pmatrix} \begin{pmatrix} P \\ Q \end{pmatrix}$$
    where $P$ is the projector onto the detected state and $Q = 1-P$.
  • The parameter $x = 1 - \sqrt{1-\eta}$ is introduced to facilitate perturbation theory, interpolating between the unitary limit ($\eta=0, x=0$) and the projective limit ($\eta=1, x=1$).

• Analytical Tools:

  • Generating Functions: The authors analyze the Fourier transforms of the return amplitudes, $\hat{\phi}_\eta(z)$, to derive exact relations between weakly monitored and projectively monitored systems.
  • Perturbation Theory: They utilize convergent perturbation expansions in powers of $x$ (near the unitary limit) and powers of $(1-x)$ (near the projective limit).
  • Spectral Analysis: The variance and moments of the return time are calculated using the spectral representation of the monitored evolution operator $Q_\eta U$.
  • Topological Analysis: The winding number is derived from the phase of the return amplitude on the unit circle in the complex plane.

3. Key Contributions

1. Scaling Relation for Mean Return Time: The paper derives a fundamental scaling law for the mean return time $\langle t \rangle$ under weak monitoring:
   $$\langle t \rangle = \tau \frac{n_w}{\eta}$$
   where $\tau$ is the time step, $n_w$ is the winding number (topological invariant), and $\eta$ is the measurement strength. This shows that the mean return time diverges as $1/\eta$ as the measurement becomes weaker, but the topological factor $n_w$ remains robust.

2. Connection to Random-Time Monitoring: The authors demonstrate that weak continuous monitoring is mathematically analogous to random-time projective monitoring. Specifically, the scaling $\langle t \rangle \propto 1/\eta$ mirrors the result for random measurements where $\langle t \rangle = \langle \tau \rangle n_w$, with $1/\eta$ playing the role of the average time interval between measurements.

3. Convergent Perturbation Theory: The work establishes that the monitored return amplitude can be expanded in a convergent geometric series. This provides a practical tool for approximating return probabilities in quantum circuits where only a few measurements occur.

4. Variance Analysis: The paper calculates the variance of the return time, showing that fluctuations diverge asymptotically as $1/\eta^2$. It also identifies "variance plateaus" in two-level systems where the variance remains constant over a wide range of parameters, except near singular points.

4. Key Results

• Quantization Robustness: The mean return time is determined solely by the winding number $n_w$ (the number of detector-visible energy levels) and the measurement strength. The detailed spectral weights or specific energy values do not alter the quantization structure, only the scaling factor.
• Probability Conservation: Despite the weak nature of the measurement, the total probability of the quantum walk returning to the initial state remains 1 ($\sum |\phi_{\eta,n}|^2 = 1$).
• Two-Level System Example:
  • For a single qubit (2-level system), explicit expressions for return amplitudes and probabilities were derived.
  • The return probabilities $|\phi_{\eta,n}|^2$ do not follow a simple scaling law with $\eta$; they exhibit non-monotonic behavior where some probabilities increase and others decrease as $\eta$ changes.
  • The variance of the return time was shown to be largely constant with respect to the coupling parameter $J\tau$ (except near singularities), suggesting a degree of stability in the fluctuations.
• Mixed States: The authors note that for mixed states, the geometric series involves matrix products rather than scalars. While winding numbers can still be defined, the mean time may not be strictly quantized if different eigenvalues possess different winding numbers.

5. Significance

• Fundamental Physics: The results bridge the gap between idealized projective measurements and realistic weak measurements in quantum systems. They confirm that topological protection (via the winding number) is a robust feature that survives the transition from strong to weak monitoring.
• Quantum Control & Diagnostics: The derived scaling laws and perturbation expansions provide practical tools for designing quantum algorithms and diagnostic protocols. Understanding how measurement strength affects return times is crucial for quantum feedback control and error correction.
• Experimental Relevance: The framework applies to various physical implementations, including superconducting quantum processors and circuit-based architectures where ancilla coupling is the standard method for measurement. The analogy to random-time monitoring suggests that weak monitoring can be used to simulate or approximate stochastic measurement protocols.
• Theoretical Framework: The paper offers a rigorous mathematical link between the unitary evolution of a quantum walk and its monitored counterpart, utilizing convergent perturbation theory to handle the interpolation between limits.

In conclusion, the paper demonstrates that while weak measurements introduce significant fluctuations and scale the mean return time inversely with measurement strength, the underlying topological quantization of the return time remains intact, governed by the winding number of the system's characteristic function.