[Paper Review] Error suppression in Hamiltonian based quantum computation using energy penalties

Adam D. Bookatz, Edward Farhi+1 more|arXiv (Cornell University)|Jul 6, 2014

Quantum Computing Algorithms and Architecture24 references45 citations

TL;DR

This paper proposes using energy penalties with quantum error-detecting codes to suppress environmentally induced errors in Hamiltonian-based quantum computation. By encoding logical qubits and adding a constant energy penalty Hamiltonian that suppresses states outside the codespace, the authors prove that in the infinite penalty limit, 1-local errors are completely suppressed, and derive finite-penalty bounds that are supported by numerical simulations showing even better protection than predicted.

ABSTRACT

We consider the use of quantum error detecting codes, together with energy penalties against leaving the codespace, as a method for suppressing environmentally induced errors in Hamiltonian based quantum computation. This method was introduced in [1] in the context of quantum adiabatic computation, but we consider it more generally. Specifically, we consider a computational Hamiltonian, which has been encoded using the logical qubits of a single-qubit error detecting code, coupled to an environment of qubits by interaction terms that act one-locally on the system. Energy penalty terms are added that penalize states outside of the codespace. We prove that in the limit of infinitely large penalties, one-local errors are completely suppressed, and we derive some bounds for the finite penalty case. Our proof technique involves exact integration of the Schrodinger equation, making no use of master equations or their assumptions. We perform long time numerical simulations on a small (one logical qubit) computational system coupled to an environment and the results suggest that the energy penalty method achieves even greater protection than our bounds indicate.

Motivation & Objective

• To develop a passive, Hamiltonian-compatible method for suppressing environmental errors in quantum computation without intermediate measurements or active corrections.
• To prove that energy penalties can suppress 1-local errors arbitrarily well in the limit of infinite penalty, using exact Schrödinger equation integration.
• To derive finite-penalty bounds for error suppression and assess their tightness via numerical simulations.
• To generalize the method to k-local errors using k-qubit error-detecting codes, showing the approach remains valid under broader error models.
• To evaluate the method's potential for fault-tolerant adiabatic quantum computation and Hamiltonian simulation.

Proposed method

• Encodes the computational Hamiltonian using a single-qubit quantum error-detecting code, mapping logical qubits to multiple physical qubits.
• Adds a constant energy penalty term to the Hamiltonian that projects out states outside the codespace, penalizing any state with a single-qubit error.
• Uses exact integration of the Schrödinger equation to analyze time evolution, avoiding approximations from master equations.
• Applies the Riemann-Lebesgue lemma to show that the error amplitude vanishes in the infinite energy penalty limit.
• Derives finite-penalty bounds by analyzing the phase evolution of error terms in the interaction picture.
• Performs long-time numerical simulations on a 1-logical-qubit system (4 physical qubits) coupled to an 8-qubit environment to test performance.

Experimental results

Research questions

• RQ1 Can energy penalties effectively suppress 1-local environmental errors in Hamiltonian-based quantum computation without active error correction?
• RQ2 What is the quantitative bound on error suppression for finite energy penalties, and how tight are these bounds?
• RQ3 Does the energy penalty method remain effective when generalized to k-local errors using a k-qubit error-detecting code?
• RQ4 How does the method perform in practice compared to theoretical bounds, particularly in small-scale simulations?
• RQ5 Can this approach provide sufficient protection for adiabatic quantum computation with realistic penalty strengths?

Key findings

• In the limit of infinite energy penalty, the method completely suppresses 1-local errors, ensuring the system evolves exactly as if no environmental interaction occurred.
• Finite-penalty bounds are derived, showing that error suppression improves with increasing penalty strength, though the bounds are conservative.
• Numerical simulations on a 1-logical-qubit system (4 physical qubits, 8 environment qubits) show that system fidelity remains nearly perfect for energy penalties EP ≥ 16.
• The observed protection in simulations exceeds the theoretical bounds, suggesting the bounds can be improved.
• The method generalizes to k-local errors when using a k-qubit error-detecting code, provided the interaction Hamiltonian satisfies PV P = 0.
• The energy penalty method is passive and compatible with the Hamiltonian model, avoiding the need for intermediate measurements or fast control pulses.