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Quantum-JumpContinuous-Time QuantumError-Correction

Kung-Chuan Hsu, Todd A. Brun

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

• Continuous-time quantum error-correction (CTQEC) is a study of pro-tecting quantum information against decoherence, where both the decoher-ence and error-correction processes are considered continuous in time. • Quantum-jump CTQEC is a model for CTQEC that describes a quantum sys-tem with a continuous master equation. One important task in concerning this model is to find a protocol that realizes the quantum-jump CTQEC mapping ρ δt1−−−→ 1 − κδt ρ+ κδt R(ρ), (1) ε ε whereR(·) is a recovery mapping. Contribution • For any given quantum stabilizer code, we found a set of criterion that leads to a large family of protocols that realize the corresponding quantum-jump CTQEC mapping. • We propose a CTQEC protocol, based on any given [[n, k, d]] quantum stabilizer code, that requires only n − k + 1 ancillary qubits in the quantum measurement phase. This greatly reduces the number of ancillary qubits required from 2n−k as in [1]. • The comparison of existing CTQEC protocols. Typical Procedures in a CTQEC protocol • Step I: Couple an ancillary system of size sa, prepared in the state |A0 〉 = |+〉⊗sa =

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Continuous-Time Quantum Error Correction via Lindblad Feedback Control — E8 Intelligence Research

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FINDING: Continuous-time quantum error correction (CTQEC) unifies noise and correction as a single Markovian open-system dynamics governed by the Lindblad master equation, enabling feedback control without discrete syndrome measurement. | MATH: dρ/dt = -i[H,ρ] + Σ_k (L_k ρ L_k† - ½{L_k†L_k, ρ}); steady-state condition: Σ_k (L_k ρ_ss L_k† - ½{L_k†L_k, ρ_ss}) = 0; code-space projection P_c with feedback Hamiltonian H_fb = Σ_a λ_a M_a (M_a = measurement operators); recovery fidelity F = Tr[P_c ρ(t)] → 1 as t→∞ under Knill-Laflamme condition P_c L_k† L_j P_c = α_kj P_c | CONNECTION: Knill-Laflamme condition defines a code subspace invariant under error operators — structurally identical to a stabilizer lattice / root-system invariant subspace; continuous weak measurement rates and feedback gains form a control manifold whose fixed points correspond to decoherence-free subspaces (DFS), which are the quantum analogue of crystallographic symmetry-protected subspaces. No direct appearance of φ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Extending quantum error correction: new continuous measurement protocols and improved fault-tolerant overhead

John P. Preskill, Charlene Ahn

Quantum mechanical applications range from quantum computers to quantum key distribution to teleportation. In these applications, quantum error correction is extremely important for protecting quantum states against decoherence. Here I present two main results regarding quantum error correction protocols. The first main topic I address is the development of continuous-time quantum error correction protocols via combination with techniques from quantum control. These protocols rely on weak measurement and Hamiltonian feedback instead of the projective measurements and unitary gates usually assumed by canonical quantum error correction. I show that a subclass of these protocols can be understood as a quantum feedback protocol, and analytically analyze the general case using the stabilizer formalism; I show that in this case perfect feedback can perfectly protect a stabilizer subspace. I also show through numerical simulations that another subclass of these protocols does better than canonical quantum error correction when the time between corrections is limited. The second main topic is development of improved overhead results for fault-tolerant computation. In particular, through analysis of topological quantum error correcting codes, it will be shown that the required blowup in depth of a noisy circuit performing a fault-tolerant computation can be reduced to a factor of O(log log L), an improvement over previous results. Showing this requires investigation into a local method of performing fault-tolerant correction on a topological code of arbitrary dimension.

•Jan 1, 2004•8 citations
• Continuous-time quantum error-correction (CTQEC) is a study of pro-tecting quantum information against decoherence, where both the decoher-ence and error-correction processes are considered continuous in time. • Quantum-jump CTQEC is a model for CTQEC that describes a quantum sys-tem with a continuous master equation. One important task in concerning this model is to find a protocol that realizes the quantum-jump CTQEC mapping ρ δt1−−−→ 1 − κδt ρ+ κδt R(ρ), (1) ε ε whereR(·) is a recovery mapping. Contribution • For any given quantum stabilizer code, we found a set of criterion that leads to a large family of protocols that realize the corresponding quantum-jump CTQEC mapping. • We propose a CTQEC protocol, based on any given [[n, k, d]] quantum stabilizer code, that requires only n − k + 1 ancillary qubits in the quantum measurement phase. This greatly reduces the number of ancillary qubits required from 2n−k as in [1]. • The comparison of existing CTQEC protocols. Typical Procedures in a CTQEC protocol • Step I: Couple an ancillary system of size sa, prepared in the state |A0 〉 = |+〉⊗sa =
Continuous-time quantum error correction (CTQEC) is a technique for protecting quantum information against decoherence, where both the decoherence and error correction processes are considered continuous in time. Given any [[n,k,d]][[n,k,d]][[n,k,d]] quantum stabilizer code, we formulate a class of protocols to implement CTQEC, involving weak coherent measurements and weak unitary corrections. Under this formalism, we show that the minimal required size of the ancillary system is $n\ensuremath{-}k+1$ qubits, and we propose one scheme that meets this minimal requirement. Furthermore, we compare our method with other known schemes, and show that a particular measure of performance described in this paper is better when using our method.
Continuous-time quantum error correction (CTQEC) is an approach to protecting quantum information from noise in which both the noise and the error correcting operations are treated as processes that are continuous in time. This chapter investigates CTQEC based on continuous weak measurements and feedback from the point of view of the subsystem principle, which states that protected quantum information is contained in a subsystem of the Hilbert space. We study how to approach the problem of constructing CTQEC protocols by looking at the evolution of the state of the system in an encoded basis in which the subsystem containing the protected information is explicit. This point of view allows us to reduce the problem to that of protecting a known state, and to design CTQEC procedures from protocols for the protection of a single qubit. We show how previously studied CTQEC schemes with both direct and indirect feedback can be obtained from strategies for the protection of a single qubit via weak measurements and weak unitary operations. We also review results on the performance of CTQEC with direct feedback in cases of Markovian and non-Markovian decoherence, where we have shown that due to the existence of a Zeno regime in non-Markovian dynamics, the performance of CTQEC can exhibit a quadratic improvement if the time resolution of the weak error-correcting operations is high enough to reveal the non-Markovian character of the noise process.
FINDING: Continuous-time quantum error correction (CTQEC) unifies noise and correction as a single Markovian open-system dynamics governed by the Lindblad master equation, enabling feedback control without discrete syndrome measurement. | MATH: dρ/dt = -i[H,ρ] + Σ_k (L_k ρ L_k† - ½{L_k†L_k, ρ}); steady-state condition: Σ_k (L_k ρ_ss L_k† - ½{L_k†L_k, ρ_ss}) = 0; code-space projection P_c with feedback Hamiltonian H_fb = Σ_a λ_a M_a (M_a = measurement operators); recovery fidelity F = Tr[P_c ρ(t)] → 1 as t→∞ under Knill-Laflamme condition P_c L_k† L_j P_c = α_kj P_c | CONNECTION: Knill-Laflamme condition defines a code subspace invariant under error operators — structurally identical to a stabilizer lattice / root-system invariant subspace; continuous weak measurement rates and feedback gains form a control manifold whose fixed points correspond to decoherence-free subspaces (DFS), which are the quantum analogue of crystallographic symmetry-protected subspaces. No direct appearance of φ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum error correction (QEC) is advancing from theoretical stabilizer codes to practical fault-tolerance thresholds, with continuous-time formulations emerging as a key frontier. MATH: - **Stabilizer formalism**: Pauli group Pn \mathcal{P}_n Pn​ on n n n qubits; code space C={∣ψ⟩:Si∣ψ⟩=∣ψ⟩ ∀i} \mathcal{C} = \{|\psi\rangle : S_i|\psi\rangle = |\psi\rangle \ \forall i\} C={∣ψ⟩:Si​∣ψ⟩=∣ψ⟩ ∀i}, with Si S_i Si​ generators of an abelian subgroup of Pn \mathcal{P}_n Pn​. - **Distance & threshold**: A [[n,k,d]][[n,k,d]][[n,k,d]] code corrects t=⌊(d−1)/2⌋ t = \lfloor (d-1)/2 \rfloor t=⌊(d−1)/2⌋ errors; fault-tolerance threshold pth≈10−3 p_{\text{th}} \approx 10^{-3} pth​≈10−3–10−210^{-2} 10−2 per gate (surface code). - **Continuous-time QEC (CTQEC)**: Master equation ρ˙=−i[H,ρ]+∑kγkD[Lk]ρ \dot{\rho} = -i[H,\rho] + \sum_k \gamma_k \mathcal{D}[L_k]\rho ρ˙​=−i[H,ρ]+∑k​γk​D[Lk​]ρ, with weak measurements and feedback; error rate scales as γk∝(measurement strength)2/noise bandwidth \gamma_k \propto (\text{measurement strength})^2 / \text{noise bandwidth} γk​∝(measurement strength)2/noise bandwidth. - **QuOps metric**: Reliable quantum operations ∼103 \sim 10^3 ∼103–10410^4 104 currently; tar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Continuous-time quantum error correction (QEC) integrates weak measurement, feedback control, and lattice gauge theory to stabilize logical qubits against decoherence, with the key mathematical structure being the stochastic master equation (SME) and its steady-state fidelity thresholds. | MATH: The core is the SME: dρ=−i[H,ρ]dt+∑kD[Lk]ρ dt+∑kηk H[Lk]ρ dWkd\rho = -i[H,\rho]dt + \sum_k \mathcal{D}[L_k]\rho\,dt + \sum_k \sqrt{\eta_k}\,\mathcal{H}[L_k]\rho\,dW_kdρ=−i[H,ρ]dt+∑k​D[Lk​]ρdt+∑k​ηk​​H[Lk​]ρdWk​, where D[L]ρ=LρL†−12{L†L,ρ}\mathcal{D}[L]\rho = L\rho L^\dagger - \frac{1}{2}\{L^\dagger L,\rho\}D[L]ρ=LρL†−21​{L†L,ρ} is the dissipator, H[L]ρ=Lρ+ρL†−Tr[(L+L†)ρ]ρ\mathcal{H}[L]\rho = L\rho + \rho L^\dagger - \mathrm{Tr}[(L+L^\dagger)\rho]\rhoH[L]ρ=Lρ+ρL†−Tr[(L+L†)ρ]ρ is the measurement back-action, and dWkdW_kdWk​ are Wiener increments. For lattice gauge theory (e.g., toric code), stabilizer operators Av=∏i∈vσixA_v = \prod_{i\in v} \sigma^x_iAv​=∏i∈v​σix​, Bp=∏i∈pσizB_p = \prod_{i\in p} \sigma^z_iBp​=∏i∈p​σiz​ define the code space; continuous measurement of these yields a feedback gain ggg with optimal fidelity scaling as F∼1−O(γ/g)F \sim 1 - O(\gamma/g)F∼1−O(γ/g) for measurement rate \(\ga Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Quantum mechanical applications range from quantum computers to quantum key distribution to teleportation. In these applications, quantum error correction is extremely important for protecting quantum states against decoherence. Here I present two main results regarding quantum error correction protocols. The first main topic I address is the development of continuous-time quantum error correction protocols via combination with techniques from quantum control. These protocols rely on weak measurement and Hamiltonian feedback instead of the projective measurements and unitary gates usually assumed by canonical quantum error correction. I show that a subclass of these protocols can be understood as a quantum feedback protocol, and analytically analyze the general case using the stabilizer formalism; I show that in this case perfect feedback can perfectly protect a stabilizer subspace. I also show through numerical simulations that another subclass of these protocols does better than canonical quantum error correction when the time between corrections is limited. The second main topic is development of improved overhead results for fault-tolerant computation. In particular, through analysis of topological quantum error correcting codes, it will be shown that the required blowup in depth of a noisy circuit performing a fault-tolerant computation can be reduced to a factor of O(log log L), an improvement over previous results. Showing this requires investigation into a local method of performing fault-tolerant correction on a topological code of arbitrary dimension.