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QEM ↔︎ QEC

This project is currently under development.

Why Algo-cooling?

To achieve physical implementation for quantum computation, we need:

  • A scalable physical system with well characterized qubits.
  • The ability to initialize the state of the qubits to a simple fiducial state, such as \(\ket{000\cdots}\).
  • Long relevant decoherence times, much longer than the gate operation time.
  • A “universal” set of quantum gates.
  • A qubit-specific measurement capability.

There are 2 methods for doing this:

  1. “natural cooling” to the ground state of the system-Hamiltonian
  2. cooling by projective measurement.

In ensemble quantum computation models (such as NMR platform),

To-dos

  • July 31, 2024 | Verify argument on algorithmic cooling for generating low entropy states, their use in QEC encoding, and potential limitations due to gate imperfections.
  • Aug 05, 2024 | Verification of statement on using algorithmic cooling to achieve low entropy states with finite temperature reservoirs, potentially limited by gate errors.
  • Sept 22, 2024 | The refrigerator is a compression algorithm that produces one "reset" qubit in a near-pure state, with an error less than \(\epsilon_2\), using \(R\) qubits. The waste qubits are nearly maximally mixed and may be entangled with the reset qubit. A refrigerator with these properties can be achieved with a constant \(R\) and a quantum circuit of constant size \(F\).
  • Oct 04, 2024 |
    • Liang: Key point --> QEC has redundancies (provided from ancilla qubits), whereas QEM does not.
    • Mid-circuit measurement is probably not a thing to worry about in terms of the amount of information-obtaining. Because in QEC, in principle we can defer all syndrome measurements to the end of the circuit, which is the same settings as in QEM protocols.
    • Liang: Even if the cooling circuits (refrigerators) are required to be in contact with heatbaths, we can include those heatbaths into the system so that the whole system will still remain a closed system.
  • Oct 14, 2024 |
    Hi Andy, ..., Since I might be busy with conferences (IMSI workshop and Chicago Quantum Summit) in the coming two weeks, it might be better that two of you start meet and discuss about quantum algorithmic cooling. For example, it would be good to think about the following:
    • What is the fundamental limit of quantum algorithmic cooling with imperfect quantum gates (without quantum encoding)?
    • Can we improve the quantum cooling at the logical level, if we are allowed to have quantum encoding?
  • How does \(F\) scale up with \(R\) and \(\epsilon_2\)?
    • i.e., what should the depth of the algo-cooling circuit be, with an input of \(R\) qubits and a final cooling goal of \(\epsilon_2\)-deviation from a perfect \(\ket0\bra0\) state?
  • Moreover, what's the dependence of \(F\) on state \(P\)?
    • \(P\) is the non-unital channel's fixed point.
    • i.e., \(P\) already gives us an equivalent of the cooling process (to some degree), now we want to know how much more cooling we can achieve with the algo-cooling circuit, or how much of the total effort for cooling is already done by \(P\).
  • If we have imperfect quantum gates, ...
  • If we are allowed to have encoding, ...

reference: [@baughExperimentalImplementationHeatbath2005; @ben-orQuantumRefrigerator2013; @laflammeAlgorithmicCoolingResolving2022; @linThermodynamicAnalysisAlgorithmic2024; @moussaHeatBathAlgorithmicCooling; @rodriguez-brionesAchievablePolarizationHeatBath2016]