From Better Entanglement Distillation to Multi-formalism Quantum Simulation Tools
Stefan KrastanovUMass Amherst
The Quantum Technology Stack
Materials
Analog Control
Noisy Digital Circuits
Error Correction
Quantum Algorithms
Optimizing Entanglement Distillation
They can be entangled!
A=∣ϕ+⟩=2∣00⟩+∣11⟩B=∣ψ−⟩=2∣01⟩−∣10⟩C=∣ψ+⟩=2∣01⟩+∣10⟩D=∣ϕ−⟩=2∣00⟩−∣11⟩
They can be entangled!
A=∣ϕ+⟩=2∣00⟩+∣11⟩B=∣ψ−⟩=2∣01⟩−∣10⟩C=∣ψ+⟩=2∣01⟩+∣10⟩D=∣ϕ−⟩=2∣00⟩−∣11⟩
- Krastanov et al.Art installation on Non-contextual Hidden Variable theories
They can be noisy!
Desired:
A=2∣00⟩+∣11⟩The hardware generated:
90% chance for A=2∣00⟩+∣11⟩ 10% chance for a bit flip on Bob's qubit C=2∣01⟩+∣10⟩Purification of entanglement
A=∣ϕ+⟩=2∣00⟩+∣11⟩B=∣ψ−⟩=2∣01⟩−∣10⟩C=∣ψ+⟩=2∣01⟩+∣10⟩D=∣ϕ−⟩=2∣00⟩−∣11⟩
A∝∣00⟩+∣11⟩B∝∣01⟩−∣10⟩C∝∣01⟩+∣10⟩D∝∣00⟩−∣11⟩
A∝∣00⟩+∣11⟩B∝∣01⟩−∣10⟩C∝∣01⟩+∣10⟩D∝∣00⟩−∣11⟩
A∝∣00⟩+∣11⟩B∝∣01⟩−∣10⟩C∝∣01⟩+∣10⟩D∝∣00⟩−∣11⟩
A convenient representation for the density matrix
A hypercube of probabilities
A∝∣00⟩+∣11⟩B∝∣01⟩−∣10⟩C∝∣01⟩+∣10⟩D∝∣00⟩−∣11⟩
What if we want multi-sacrifice circuit?
Discrete Optimization: Evolutionary Algorithm
Discrete Optimization: Evolutionary Algorithm
Discrete Optimization: Evolutionary Algorithm
- Krastanov et al.Optimized Entanglement Purification
- Jansen et al.Enumerating all bilocal Clifford distillation protocols through symmetry reduction
n-to-k purification protocols¹
- Vaishnavi Addala, Stefan Krastanov
Optimization of concatenated
purification ⟶ ECC teleportation
protocols¹
- Vaishnavi Addala, Stefan Krastanov
Modeling Entanglement More Efficiently¹
- Shu Ge, Stefan Krastanov
Error-detection circuits are Clifford circuits
Clifford circuits are efficient to simulate on stabilizer states
A∝∣00⟩+∣11⟩∼[++XXZZ]B∝∣01⟩−∣10⟩∼[−−XXZZ]C∝∣01⟩+∣10⟩∼[−+XXZZ]D∝∣00⟩−∣11⟩∼[+−XXZZ]
A⊗B⊗D∼++−−+−XXZZIIIIIIIIIIIIXXZZIIIIIIIIIIIIXXZZ
A⊗B⊗D∼++−−+−XXZZXXZZXXZZ
Bell state tableaux are block-diagonal and error-detection circuits only permute the Bell basis!
A⊗B⊗D∼++−−+−XXXXXX
Bell state tableaux are block-diagonal and error-detection circuits only permute the Bell basis!
n pairsA⊗B…
010⋮0⎭⎬⎫O(2n)
++XXZZ⋱⎭⎬⎫O(n×n)
2n bits0011…
- Let us call them "Bell Permuting" or "Bell Preserving" gates
- Let us denote them BP(n)
- Permuting gates are Clifford gates
- Not all permutations are local
- Are there Non-Clifford gates that are useful for distillation?
All Clifford gates Cn on n qubits
∣Cn∣=2(4n−1)4n∣Cn−1∣
Clifford gates Cn∗ that do not specify phases
∣Cn∗∣=4n1∣Cn∣
i.e. Cn∗=Cn/Pn
i.e. any Cn gate can be written as
C.P ∣ P∈Pn,C∈Cn∗
i.e. any Cn gate can be written as
C.P ∣ P∈Pn,C∈Cn∗
Bn
Bn∗=Bn/Pn=Sp(2n,F2)
B2∗=Sp(4,F2)={Aliceg⊗Bobg ∣ g∈C2∗}12
a useful set of cosets: Q=C2∗/(C1∗⊗C1∗)
which provides 20 "distinct" two-qubit gates
which provides 20 "distinct" two-qubit gates
- Jansen et al.Enumerating all bilocal Clifford distillation protocols through symmetry reduction
- Dehaene et al.Local permutations of products of Bell states and entanglement distillation
Bell Preserving (bi-local) gates on 2 Bell pairs
A B2 gate can be written as Alice(g.p)⊗Bob(g)
where p∈P2
where p∈P2
and g∈C2∗ can be written as g=g0.(h⊗f)
h∈C1∗ and f∈C1∗ and g0∈Q
h∈C1∗ and f∈C1∗ and g0∈Q
Bell Preserving (bi-local) gates on 2 Bell pairs
p1∈P1 p2∈P1
2 Pauli gates(4 possibilities each)
h∈C1∗ f∈C1∗
2 single-qubit Cliffords(6 possibilities each)
g0∈Q=C2∗/(C1∗⊗C1∗)
20 "inherently multiqubit" gates20×6×6×4×4 = 720×16 = 11520 different Bell Preserving 2-pair gates
Bell Preserving (bi-local) gates
That also maps AA ⟶ AAThere 720 such gates
But just 36 suffice for state of the art purification...
There are 6 possible options for ha⊗hb and fa⊗fb each.
Namely the 6 permutations of BCD.
Namely the 6 permutations of BCD.
Taking Optimization Seriously
Even your Monte Carlo simulations should be "differentiable"!¹- Arya et al.Automatic Differentiation of Programs with Discrete Randomness
Full-Stack Design and Optimization Toolkit
Types of Dynamics
Types of Dynamics
Hamiltonians, Master Equations
Types of Dynamics
Hamiltonians, Master Equations
Gates, Circuits
Types of Dynamics
Hamiltonians, Master Equations
Gates, Circuits
Weak Measurements, Feedback
State Representation
traits = [Qubit(), Qubit(), Qumode()]
reg = Register(traits)
A register "stores" the states being simulated.
graph = grid([2,3])
registers = [...]
net = RegisterNet(graph, registers)
A "graph" of registers can represent a network.
initialize!(reg[1], X₁)
A register's slot can be initialized to an arbitrary state, e.g. ∣x1⟩ an eigenstate of σ^x.
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
project_traceout!(reg[1], σˣ) # Projective measurement observable((reg[1],reg[2]), σᶻ⊗σˣ) # Calculate an expectation
Measurements and expectation values...
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 + 0.0imjulia> express( Y₁⊗Y₂, CliffordRepr() )
Rank 2 stabilizer
+ Z_
+ _Z
════
+ Y_
- _Y
════Play with it at areweentangledyet.com
Other features...
Declarative specification of "imperfections"
Discrete event scheduling
Traveling wavepackets modeling
More formalisms
More symbolic algebra
Digital twin / surrogate modeling
QuantumSavory.jl
github.com/Krastanov/QuantumSavory.jl
Better modeling and better optimization for entanglement exist
Try out QuantumClifford.jl - it is public and stable
Be an early tester for QuantumSavory.jl
Including work done by Vaishnavi Addala and Shu Ge, in coordination with Dirk Englund.
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.
Creating new tools for the entire community.
From Better Entanglement Distillation to Multi-formalism Quantum Simulation Tools Stefan Krastanov UMass Amherst Dec 2022