Continuous measurement-based holonomic quantum computation

Abstract

We propose a scheme to generate holonomies using the Quantum Zeno effect, enabling logical unitary operations on quantum stabilizer codes purely through measurements. The quantum error-correcting code space is adiabatically rotated by measuring a succession of rotated stabilizer generators. When the rotation is sufficiently slow, the state remains confined to the instantaneous code space by the Zeno effect; otherwise, measurement-induced jumps can occur into a rotated orthogonal subspace. If the rotation completes a closed loop, the code state is transformed by a holonomy: a logical unitary transformation. We analytically derive the sequence of rotated stabilizer generators that produce a desired holonomy, and find the total time required to implement this procedure with a given success probability. If a measurement moves the state to the orthogonal subspace, we present a method to alter the path of the rotated observables to return the state either to the original code or the original error space with the desired holonomy; in the latter case, the holonomy is emulated. Finally, we establish conditions on the code and the measured observables that preserve the correctability of a given error set. When a code fails to meet the error-correcting conditions, our protocol remains applicable by augmenting the code with at most two ancilla qubits.

Figures (5)

• Figure 1: The horizontal lift as a unique curve in $S_{N, K}(\mathbb{C})$ with the base manifold $G_{N, K}(\mathbb{C})$. The difference between the initial point $V(0)$ and the final point $V(T)$ is the holonomy.
• Figure 2: Jump into the rotated error space. When the code space is not rotated adiabatically, the state can jump into an error space defined by the operator $X$. The error state can potentially include a logical error.
• Figure 3: Real-time error correction by dynamic path modulation. When a non-adiabatic evolution results in a measurement-induced error, it can be corrected by changing the holonomic path to either $(W, 7\pi/2-\zeta)$ or $(W, 4\pi-\zeta)$. In the former case, the state returns to the original code space with the desired holonomy, denoted by the green curves. In the latter case, the state returns to the original error space with the emulated holonomy, denoted by the blue curves. The operator $X$ can be stored in a Pauli frame and the subsequent computation can be performed.
• Figure 4: Probability of no fault at the end of the evolution as function of the rotation increment $\delta\phi$ for the $[\![3, 1, 3]\!]$ bit-flip code with $H=\otimes_{i=1}^3\sigma_i^x$, $\theta=\pi/6$ and $X = \sigma_1^x\sigma_3^z$. The plots are averaged over an ensemble of $2000$ trajectories. The error bars represent a $95\%$ confidence interval.
• Figure 5: Probability of a jump as a function of $\omega/\kappa$. The analytical equation is \ref{['eq:confinement_prob']}.

Theorems & Definitions (35)

• Proposition 1
• proof
• Definition 1: Holonomic path
• Proposition 2
• proof
• Lemma 1
• proof
• Theorem 1
• proof
• Example 1