From Better Entanglement Distillation to Multi-formalism Quantum Simulation Tools

Stefan Krastanov
UMass Amherst

Optimizing Entanglement Distillation

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. Addala, Ge, Krastanov
    Faster-than-Clifford Simulations of Entanglement Purification Circuits and Their Full-stack Optimization
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. Addala, Ge, Krastanov
    Faster-than-Clifford Simulations of Entanglement Purification Circuits and Their Full-stack Optimization
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. Addala, Ge, Krastanov
    Faster-than-Clifford Simulations of Entanglement Purification Circuits and Their Full-stack Optimization

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!

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

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

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

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

What (bi-local) gates exist in this more efficient formalism?
  • Let us call them "Bell Permuting" or "Bell Preserving" gates
  • Let us denote them $\mathrm{BP}(n)$
  • Permuting gates are Clifford gates
  • Not all permutations are local
  • Are there Non-Clifford gates that are useful for distillation?
The most general Bell Permuting gate?
All Clifford gates $\mathcal{C}_n$ on n qubits $$ |\mathcal{C}_n| = 2(4^n-1)4^n|\mathcal{C}_{n-1}| $$
Clifford gates $\mathcal{C}^*_n$ that do not specify phases $$ |\mathcal{C}^*_n| = \frac{1}{4^n}|\mathcal{C}_n| $$
i.e. $\mathcal{C}^*_n = \mathcal{C}_n/\mathcal{P}_n$
i.e. any $\mathcal{C}_n$ gate can be written as
$C.P\ |\ P\in\mathcal{P}_n\,,C\in\mathcal{C}^*_n$
Bell Preserving (bi-local) gates on n Bell pairs
$$ \mathcal{B}_n $$
$$ \mathcal{B}^*_n = \mathcal{B}_n / \mathcal{P}_n = \mathrm{Sp}(2n, \mathbb{F}_2)$$
$$ \mathcal{B}^*_2 = \mathrm{Sp}(4, \mathbb{F}_2) = \{\underbrace{g}_{\tiny Alice}\otimes \underbrace{g}_{\tiny Bob}\ |\ g\in\mathcal{C}^*_2\}¹²$$
a useful set of cosets: $ Q = \mathcal{C}^*_2 / (\mathcal{C}^*_1 \otimes \mathcal{C}^*_1) $
which provides 20 "distinct" two-qubit gates
  1. Jansen et al.
    Enumerating all bilocal Clifford distillation protocols through symmetry reduction
  2. Dehaene et al.
    Local permutations of products of Bell states and entanglement distillation

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

A $\mathcal{B}_2$ gate can be written as $\underbrace{(g.p)}_{\tiny Alice}\otimes\underbrace{(g)}_{\tiny Bob}$
where $p\in\mathcal{P}_2$
and $g\in\mathcal{C}^*_2$ can be written as $g = g_0.(h\otimes f)$
$h\in\mathcal{C}^*_1$ and $f\in\mathcal{C}^*_1$ and $g_0\in Q$

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
 
 

Bell Preserving (bi-local) gates

That also maps AA ⟶ AA
There 720 such gates
But just 36 suffice for state of the art purification...
There are 6 possible options for $h_a\otimes h_b$ and $f_a\otimes f_b$ each.
Namely the 6 permutations of BCD.
 
Time to perform a pair of CNOT gates, depending on formalism
  1. Addala, Ge, Krastanov
    Faster-than-Clifford Simulations of Entanglement Purification Circuits and Their Full-stack Optimization

The Quantum Technology Stack

Materials

Analog Control

Noisy Digital Circuits

Error Correction

Quantum Algorithms

Full-Stack Design and Optimization Toolkit

Symbolic description of quantum logic

Declarative noise models

Translation to many simulator backends

Discrete event scheduler

High-level lego-like interface

Symbolic description of quantum logic

Build upon Symbolics.jl and many "backend" libraries.

Full Symbolic Computer Algebra System


              julia> Z₁
              |Z₁⟩
            

              julia> ( Z₁⊗X₂+Y₁⊗Y₁ ) / √2
              0.707 (|Y₁⟩|Y₁⟩+|Z₁⟩|X₂⟩)

Symbolic to Numeric Conversion


              julia> express( ( Z₁⊗X₂+Y₁⊗Y₁ ) / √2 )
              Ket(dim=4)
                basis: [Spin(1/2) ⊗ Spin(1/2)]
                 0.8535533905932736 + 0.0im
                                0.0 + 0.3535533905932737im
               -0.49999999999999994 + 0.3535533905932737im
                -0.3535533905932737 + 0.0im
            

              julia> express( Y₁⊗Y₂, CliffordRepr() )
              Rank 2 stabilizer
              + Z_
              + _Z
              ════
              + Y_
              - _Y
              ════

The express interface


              express(::Symbolic, ::Representation, ::Use)
              express_nolookup(::Symbolic, ::Representation, ::Use)

Translation to many simulator backends


                traits = [Qubit(), Qubit(), Qumode()]
                reg = Register(traits)

A register "stores" the states being simulated.


                initialize!(reg[1], X₁)

A register's slot can be initialized to an arbitrary state, e.g. $|x_1\rangle$ an eigenstate of $\hat{\sigma}_x$.


                initialize!(reg[1], X₁)
                initialize!(reg[2], Z₁)
                apply!((reg[1], reg[2]), CNOT)

Arbitrary quantum gates or channels can be applied.


                initialize!(reg[1], X₁)
                initialize!(reg[2], Z₁)
                apply!((reg[1], reg[2]), CNOT)

Arbitrary quantum gates or channels can be applied.


              project_traceout!(reg[1], σˣ) # Projective measurement

              observable((reg[1],reg[2]), σᶻ⊗σˣ) # Calculate an expectation

Measurements and expectation values...

Automatic tracking of noise processes


                reg = Register([Qubit(), Qubit(), Qubit()])
              

                reg = Register(
                  [Qubit(), Qubit(), Qubit()]
                  [T1Decay(T₁), CoherentError(ε*σᶻ), NZ(...)]
                )
              

                reg = Register(
                  [Qubit(), Qubit(), Qubit()]
                  [T1Decay(T₁), CoherentError(ε*σᶻ), NZ(...)]
                )
                apply!((reg[1],reg[2]), CNOT; time=τ₁)
              

                reg = Register(
                  [Qubit(), Qubit(), Qubit()]
                  [T1Decay(T₁), CoherentError(ε*σᶻ), NZ(...)]
                )
                apply!((reg[1],reg[2]), CNOT; time=τ₁)
                apply!((reg[2],reg[3]), CPHASE; time=τ₂)

Discrete event scheduler

Locks and channels, message passing, delays, concurrency, agent-based sims...

Discrete event scheduler

High-level lego-like interface

Full repeater sim with automatic instrumentation.

QuantumSavory.jl

github.com/QuantumSavory

prerelease...

A few state-of-the-art Simulators

The wider Julia ecosystem

QuantumOptics.jl, ITensors.jl, Yao.jl, quantum chemistry and solid state tools, EM tools, multiphisics classical tools, etc

Most sophisticated Clifford algebra simulator

github.com/QuantumSavory/QuantumClifford.jl Multiplying two 1 gigaqubit Paulis in 32 ms.

With upcoming "Google Summer of Code" contributors working on GPU acceleration and ECC zoo.

MIT and UMass students working on code generators.

Incoming master student working on code decoders.

github.com/QuantumSavory/QuantumClifford.jl

As an "algebra" tool: random states, gate enumeration, canonicalization, partial traces, projections.

QuantumClifford.jl 

              random_stabilizer(5,10) |> 
                  canonicalize_clip! |>
                  naive_syndrome_circuit
QuantumClifford.jl

As an "circuit simulation" tool: Monte Carlo, Pauli frames, and perturbative expansions.

QuantumClifford.jl

              Dict with 3 entries:
                failure       => 4e*((1 - 3e)^3)
                false_success => 6e*((1 - 3e)^3)
                true_success  => (1 - 3e)^4 + 2e*((1 - 3e)^3)
pluto.krastanov.org/basic-purification-circuits.html
QuantumClifford.jl

support for non-Clifford expansions, conversion to QuantumOptics.jl objects and more...

Faster-than-Clifford Bell Pair circuits

github.com/QuantumSavory/BPGates.jl
Time to perform a pair of CNOT gates, depending on formalism

Waveguide Quantum Electrodynamics

github.com/qojulia/WaveguideQED.jl
Quantum wavepacket reflected from a cavity

Taking Optimization Seriously

Even your Monte Carlo simulations should be "differentiable"!

QuantumInterface.jl

github.com/qojulia/QuantumInterface.jl

We are hiring both at UMass Amherst and MIT:
software engineering
quantum science

Message at stefankr@mit.edu, skrastanov@umass.edu

 

Including work done with Vaishnavi Addala, Shu Ge, Shayan Pardis, Chen Zhao, Hong-Ye Hu, Dirk Englund, Saikat Guha.

Consider gradschool or postdoc at UMass Amherst:

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

Working on practical LDPC ECC in networking and computing with CQN.

Creating new tools for the entire community.