Noisy Entanglement (on Color Centers)

Stefan Krastanov | MIT

Entanglement

Alice has a qubit
Alice has a qubit
Bob too
They can be entangled! \[\begin{aligned} A=|\phi_{+}\rangle=\frac{|00\rangle+|11\rangle}{\sqrt{2}}\\ B=|\psi_{-}\rangle=\frac{|01\rangle-|10\rangle}{\sqrt{2}}\\ C=|\psi_{+}\rangle=\frac{|01\rangle+|10\rangle}{\sqrt{2}}\\ D=|\phi_{-}\rangle=\frac{|00\rangle-|11\rangle}{\sqrt{2}} \end{aligned}\]
They can be entangled! \[\begin{aligned} A=|\phi_{+}\rangle=\frac{|00\rangle+|11\rangle}{\sqrt{2}}\\ B=|\psi_{-}\rangle=\frac{|01\rangle-|10\rangle}{\sqrt{2}}\\ C=|\psi_{+}\rangle=\frac{|01\rangle+|10\rangle}{\sqrt{2}}\\ D=|\phi_{-}\rangle=\frac{|00\rangle-|11\rangle}{\sqrt{2}} \end{aligned}\]
Hidden Variable theories¹
  1. Krastanov et al.
    Art installation on Non-contextual Hidden Variable theories

Entanglement with Color Centers

The Color Center

(Optical) Electronic Spin for Networking
(Microwave) Nuclear Spin for Memory

Entanglement Protocols

Inherently low probability of success (DLCZ-like)

Full excitation and path erasure (Barret-Kok)

Single-photon reflection

Atom-photon gates (Duan-Kimble)

Low-probability of success codes (DLCZ-like)

Full excitation and path erasure (Barret-Kok)

Single-photon reflection

Atom-photon gates (Duan-Kimble)

Rate only classical light no optical excitation
DLCZ \( \eta (1-F) \) ✅ ❌
BK \( \eta^2 \) ✅ ❌
reflection \( \eta^2 \) ❌ ✅

You also need a color center that does not suffer from spectral diffusion or charge state instabilities.

If you end up with significant optical excitation, you also need a way to turn off the hyperfine coupling to the (nuclear) memory.

First Steps to Entanglement Purification

They can be noisy!

Desired:

\[\begin{aligned} A=\frac{|00\rangle+|11\rangle}{\sqrt{2}}\\ \end{aligned}\]

The hardware generated:

90% chance for \[\begin{aligned} A=\frac{|00\rangle+|11\rangle}{\sqrt{2}}\\ \end{aligned}\] 10% chance for a bit flip on Bob's qubit \[\begin{aligned} C=\frac{|01\rangle+|10\rangle}{\sqrt{2}}\\ \end{aligned}\]

Purification of entanglement

\[\begin{aligned} A=|\phi_{+}\rangle=\frac{|00\rangle+|11\rangle}{\sqrt{2}}\\ B=|\psi_{-}\rangle=\frac{|01\rangle-|10\rangle}{\sqrt{2}}\\ C=|\psi_{+}\rangle=\frac{|01\rangle+|10\rangle}{\sqrt{2}}\\ D=|\phi_{-}\rangle=\frac{|00\rangle-|11\rangle}{\sqrt{2}} \end{aligned}\]
Simple entanglement purification circuit
\[\begin{aligned} A\propto|00\rangle+|11\rangle\\ B\propto|01\rangle-|10\rangle\\ C\propto|01\rangle+|10\rangle\\ D\propto|00\rangle-|11\rangle \end{aligned}\]
Purification as error propagation and detection
\[\begin{aligned} A\propto|00\rangle+|11\rangle\\ B\propto|01\rangle-|10\rangle\\ C\propto|01\rangle+|10\rangle\\ D\propto|00\rangle-|11\rangle \end{aligned}\]
Purification as reshuffling of probabilities
\[\begin{aligned} A\propto|00\rangle+|11\rangle\\ B\propto|01\rangle-|10\rangle\\ C\propto|01\rangle+|10\rangle\\ D\propto|00\rangle-|11\rangle \end{aligned}\]
Purification as reshuffling of probabilities
Possible coincidence measurements

... How are we going to find the "good" permutation?
What if we want multi-sacrifice circuit?

Discrete Optimization: Evolutionary Algorithm

Discrete Optimization: Evolutionary Algorithm

Mutations

Discrete Optimization: Evolutionary Algorithm

Mutations
Offspring circuits
(a) best previously known circuit (b) our optimized circuit¹
(c) resource usage for each
  1. Krastanov et al.
    Optimized Entanglement Purification
A collection of optimized circuits at qevo.krastanov.org¹
  1. Krastanov et al.
    Optimized Entanglement Purification
Optimized circuits for purification¹, better than any known circuits.
  1. Krastanov et al.
    Optimized Entanglement Purification

Circuit Constraints

In some hardware types only one of the qubits can establish entanglement.

Ongoing projects and useful tools

Faster purificaiton simulators¹

n-to-k purification circuits²

Rigorous theory bounds on purification performance³

QuantumClifford.jl and QuantumSavory.jl

  1. Shu Ge (an undergrad under advisement)
  2. Vaishnavi Addala (an undergrad under advisement)
  3. friends at Delft