[Paper Review] Holonomic quantum computing with cat-codes

Victor V. Albert, Stefan Krastanov+6 more|arXiv (Cornell University)|Feb 28, 2015

Quantum Computing Algorithms and Architecture1 references3 citations

TL;DR

This paper proposes a universal holonomic quantum computing framework using cat-codes—superpositions of d well-separated coherent states—via two adiabatic gate families: (1) geometric Berry phase accumulation by adiabatically encircling a coherent state in phase space, and (2) coherent population transfer through controlled collision of two coherent states. The approach enables universal quantum computation using tunable nonlinearities in systems like trapped ions and circuit QED.

ABSTRACT

Universal quantum computation on a space consisting of superpositions of $d$ well-separated coherent states can be achieved by two families of adiabatic holonomic gates. The first gate consists of moving a coherent state around a closed path in phase space, resulting in a relative Berry phase between that state and the other states. The second gate consists of colliding two coherent states, resulting in coherent population transfer between them. Such gates should be realizable via reservoir engineering of systems which support tunable nonlinearities, such as trapped ions and circuit QED.

Motivation & Objective

• To develop a universal quantum computation framework based on cat-codes, which are superpositions of multiple well-separated coherent states.
• To address the challenge of fault-tolerant quantum computation by leveraging topological robustness of holonomic gates.
• To design two families of adiabatic holonomic gates—geometric phase accumulation and coherent state collision—for universal gate sets.
• To enable implementation in physical platforms with tunable nonlinearities, such as trapped ions and circuit QED.

Proposed method

• Implement holonomic quantum gates using adiabatic evolution in the Hilbert space of cat-codes, where the system evolves along closed paths in phase space.
• Utilize the geometric Berry phase acquired during adiabatic encircling of a coherent state to implement a controlled relative phase gate.
• Engineer coherent population transfer between two coherent states via adiabatic collision processes, enabling entangling operations.
• Leverage reservoir engineering to stabilize and control the dynamics of cat-codes in systems with tunable nonlinearities.
• Ensure fault tolerance by encoding logical qubits in superpositions of multiple coherent states, exploiting their inherent robustness to certain decoherence channels.
• Design gate sequences that are robust against certain parameter fluctuations due to the geometric nature of the holonomy.

Experimental results

Research questions

• RQ1 Can universal quantum computation be achieved using only adiabatic holonomic gates in a cat-code Hilbert space?
• RQ2 How can geometric Berry phases be generated and controlled in superpositions of coherent states?
• RQ3 What dynamical processes enable coherent population transfer between distinct coherent states in a cat-code system?
• RQ4 Can such gates be implemented in realistic platforms with tunable nonlinearities, such as circuit QED and trapped ions?
• RQ5 What are the robustness properties of these holonomic gates against control errors and decoherence?

Key findings

• Universal quantum computation is achievable using two families of adiabatic holonomic gates in a cat-code framework.
• Adiabatic encircling of a coherent state in phase space generates a relative Berry phase, enabling single-qubit holonomic gates.
• Controlled collision of two coherent states enables coherent population transfer, realizing entangling two-level operations.
• The proposed gates are robust against certain control errors due to their geometric, holonomic nature.
• The framework is realizable in physical systems with tunable nonlinearities, such as trapped ions and circuit QED, via reservoir engineering.
• The approach provides a path to fault-tolerant quantum computation using macroscopic superpositions with inherent error resilience.