Better Entanglement Distillation

Stefan Krastanov | UMass Amherst
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

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

A convenient representation for the density matrix

A hypercube of probabilities
\[\begin{aligned} A\propto|00\rangle+|11\rangle\\ B\propto|01\rangle-|10\rangle\\ C\propto|01\rangle+|10\rangle\\ D\propto|00\rangle-|11\rangle \end{aligned}\]
... 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
Optimized circuits for purification¹, better than any known circuits.
  1. Krastanov et al.
    Optimized Entanglement Purification
  2. Jansen et al.
    Enumerating all bilocal Clifford distillation protocols through symmetry reduction

n-to-k purification protocols¹

  1. Shu Ge, Vaishnavi Addala, Stefan Krastanov
Purification protocols with varying sacrificial pairs, purified pairs, and register width.
A minimum-width register is necessary. Here, with n-to-2 purification examples, we need a width of 4.
A typical 5-to-2 purification circuit on a small register.
What figure of merit should we be using?
The hypercube of probabilities describing the state of 3 noisy Bell pairs.

Optimization of concatenated
purification ⟶ ECC teleportation
protocols¹

  1. Shu Ge, Vaishnavi Addala, Stefan Krastanov
For teleportation without ECC.
For distance 3 codes (1 correctable error)
For distance 5 codes (2 correctable errors)
14-to-11 purification followed by a teleportation of a [[11,2,5]] code. Each line is a differently optimized circuit.

Modeling Entanglement More Efficiently¹

  1. Shu Ge, Vaishnavi Addala, Stefan Krastanov

Error-detection circuits are Clifford circuits

Clifford circuits are efficient to simulate on stabilizer states
\[\begin{aligned} A\propto|00\rangle+|11\rangle \sim\begin{bmatrix}+&XX\\+&ZZ\end{bmatrix}\\ B\propto|01\rangle-|10\rangle \sim\begin{bmatrix}-&XX\\-&ZZ\end{bmatrix}\\ C\propto|01\rangle+|10\rangle \sim\begin{bmatrix}-&XX\\+&ZZ\end{bmatrix}\\ D\propto|00\rangle-|11\rangle \sim\begin{bmatrix}+&XX\\-&ZZ\end{bmatrix} \end{aligned}\]
\[A\otimes B\otimes D \sim\begin{bmatrix}+&XX&II&II\\+&ZZ&II&II\\ -&II&XX&II\\-&II&ZZ&II\\ +&II&II&XX\\-&II&II&ZZ\end{bmatrix}\]
\[A\otimes B\otimes D \sim\begin{bmatrix}+&XX&&\\+&ZZ&&\\ -&&XX&\\-&&ZZ&\\ +&&&XX\\-&&&ZZ\end{bmatrix}\]
Bell state tableaux are block-diagonal and error-detection circuits only permute the Bell basis!
\[A\otimes B\otimes D \sim\begin{matrix}+&\phantom{XX}&\phantom{XX}&\phantom{XX}\\+&&&\\ -&&&\\-&&&\\ +&&&\\-&&&\end{matrix}\]
Bell state tableaux are block-diagonal and error-detection circuits only permute the Bell basis!

\[ \underbrace{A\otimes B\dots}_{n\text{ pairs}} \]

\[ \left.\begin{pmatrix}0\\1\\0\\\vdots\\0\end{pmatrix}\right\}\scriptsize \mathcal{O}(2^n) \]

\[ \left.\begin{bmatrix}+&XX&\\+&ZZ&\\&&\ddots\end{bmatrix}\right\}\scriptsize \mathcal{O}(n\times n) \]

\[ \underbrace{0011\dots}_{2n \text{ bits}} \]

Bell Preserving (bi-local) gates on 2 Bell pairs

\[ p_1 \in \mathcal{P}_1 \] \[ p_2 \in \mathcal{P}_1 \]
2 Pauli gates
(4 possibilities each)
\[ h\in\mathcal{C}^*_1 \] \[ f\in\mathcal{C}^*_1 \]
2 single-qubit Cliffords
(6 possibilities each)
\[ g_0 \in Q = \mathcal{C}^*_2 / (\mathcal{C}^*_1\otimes\mathcal{C}^*_1) \]
20 "inherently multiqubit" gates
 
20×6×6×4×4 = 720×16 = 11520 different Bell Preserving 2-pair gates
 
 
Time to perform a pair of CNOT gates, depending on formalism

Preprint arxiv:2307.0635 "Faster-than-Clifford Simulations of Entanglement Purification Circuits and Their Full-stack Optimization"

Better modeling and better optimization for entanglement exist!

Try out BPGates.jl (and the QuantumClifford.jl package)

 

Including work done by Vaishnavi Addala and Shu Ge at MIT's QPG.

Consider gradschool or postdoc at UMass Amherst:

Design of optical/mechanical/spin devices with CQN, Sandia, Mitre, and MIT.

Working on practical LDPC ECC in networking and computing.

Creating new tools for the entire community.