Faster Simulation of Quantum Entanglement with BPGates.jl

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
\[\begin{aligned} A\propto|00\rangle+|11\rangle\\ B\propto|01\rangle-|10\rangle\\ C\propto|01\rangle+|10\rangle\\ D\propto|00\rangle-|11\rangle \end{aligned}\]

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.

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.