siZZle & DBAC
(Double-Bracket Algorithmic Cooling)

Kay Giang - NTU Singapore

slides.com/khanhuyengiang

DBAC Summary

QITE formula

|\Psi(\tau)\rangle = \frac{e^{-\tau \hat{H}}}{\|e^{-\tau \hat{H}}|\Psi(0)\rangle\|} |\Psi(0)\rangle

\(\Psi(0)\): Initial state

\(\Psi(\tau)\): State at time \(\tau\)

\(\hat H\): Diagonalised Hamiltonian

Cool the initial state \(\Psi(0)\) with respect to the Hamiltonian \(\hat H\)

Recursion step

\ket{\Psi(t)}\approx e^{t[\Psi(0),\hat H]}\ket{\Psi_0}

For short duration t:

\partial_\tau {|\Psi(\tau) \rangle} = [\Psi(\tau),\hat{H}] |\Psi(\tau) \rangle\

This motivates defining the recursion step:

\ket{\psi_{k+1}} = e^{t_k[\psi_k,\hat H]} \ket{\psi_k}

\(\ket{\psi_k}\): State at step \(k\) 

DB-QITE recursion formula

\boxed{\ket{\psi_{k+1}} \approx e^{i\sqrt{t_k}\hat H} e^{i\sqrt{t_k}\psi_k} e^{-i\sqrt{t_k}\hat H}\ket{\psi_k}}
e^{t[\psi,\hat H]}|\psi\rangle = e^{i\sqrt{t}\hat H}e^{i\sqrt{t}\psi} e^{-i\sqrt{t}\hat H} e^{-i\sqrt{t}\psi}|\psi\rangle + \mathcal O(t^{3/2})

Using the group commutator relation:

DB-QITE recursion formula:

Density matrix exponentiation (DME)

DB-QITE Performance

\boxed{\ket{\psi_{k+1}} \approx e^{i\sqrt{t_k}\hat H} e^{i\sqrt{t_k}\psi_k} e^{-i\sqrt{t_k}\hat H}\ket{\psi_k}}

If we have ideal DME

\(e^{i\sqrt{t_k}\psi_k}\)

Data Taking

Part 1: siZZle and DME

Gate defintion

ZZ PTM

DME PTM

  • DME\((\pi/2)\) is also just a full SWAP
  • "Trotterization": \(N\times\)DME\((t/N) \to e^{i|\psi\rangle\langle\psi|}\) 

Sanity check: Compare DME

Yvonne Gao Nature paper

Our DME simulation

Data Taking

Part 2: Circuits PTM

Circuits PTM

3 cases to take data:

  • A - 2 qubits
  • B - 3 qubits
  • C - 4 qubits
R_Z(\phi) = e^{-i \phi Z} = \begin{pmatrix} e^{-i\phi} & 0 \\ 0 & e^{i\phi} \end{pmatrix}

Step size  

Circuit A and C

Circuit B

Circuit A PTM

Part of circuit B PTM

Circuit C
(no need to do because it is Circuit A)

This is the first recursive use of DME. 

See Quantum Dynamic Programming | Phys. Rev. Lett.

Data Taking

Part 3: Extended circuits PTM (Main result)

Main Result
(Energy drop of DB-QITE)

Take data on a grid of angles to reproduce this plot

Main Result
(Energy drop of DB-QITE)

Same as previous plot but with initial fidelity instead of angle

From the previous plot, I will calculate this plot. This is our main result.

Circuit A'
(Circuit A with qubit initialisation)

Circuit A'

Part of circuit B'
(similar to A' but with DME\((\pi/8)\))

Part of circuit B'

Bloch Sphere

How to benchmark DME

is the hardware definition of

2 approaches:

(Hardware focus) Compare to state of the art (Kjaegaard) who do QME
Problem: Kjaegaard has 2 qubits in 3D cavity with great fidelity \(\to\) they can do 30 CNOTs 

(Alternatively) Comparing 2 times DME\((\pi/2)\) against DME\((\pi)\)

Algorithmic Cooling

What is algorithmic cooling (AC)?

  • Goal: Reduce entropy computational qubits.

  • Method: Iteratively apply a unitary entropy compression operation, UUU on all qubits, which is global and complicated

  • Process:

    1. Redistributes entropy across all qubits.

    2. Pushes entropy onto mmm reset-helper qubits.

    3. Leaves computational (data) qubits colder, reset-helper qubits hotter.

Heat Bath Algorithmic Cooling

  • Similar to Algorithmic Cooling
  • BUT the Reset-helper qubit can come into contact with the bath and rethermalize (ie reset itself)
  • The popular, well-known method

Q1: reset qubit

  1. Compress entropy of Q2 into Q1
  2. Q1 came in contact with bath
  3. Compress entropy of Q3 into Q1, Q2
  4. Q1 came in contact with bath
  5. Compress entropy of Q2 into Q1

HBAC vs DB-RESET

  1. Cool state towards eigenstate
  1. Cool state towards ground state

HBAC

DB-RESET

Heat bath algorithmic cooling

  1. Requires a bath
  2. Complicated compilation of U
  3. Using a full SWAP operation
  • 3 qubits: require 2 full SWAP, 4 CNOT, 1 Toffoli
  1. Requires no bath
  2. Simple decomposition
  3. Partial SWAP using hardware-natural decomposition \(\to\) faster:
  • 3 qubits: require 3 full ZZ gate

HBAC vs DB-RESET

HBAC

DB-RESET

Heat bath algorithmic cooling

  1. Works on any state
  2. Cools an ensemble of qubits

  3. Can operate on mixed states, needs dephasing 

  1. Only works on pure states
  2. Cools one qubit
  3. Doesn't need dephasing, can operate on state with coherence but must be pure

HBAC vs DB-RESET

Both: Building a hierachy of baths that gets colder

HBAC

DB-RESET

Heat bath algorithmic cooling

DBAC on mixed state

DBAC on mixed state

Even if the state is not arbitrarily pure, DBAC can reduce energy

DB-QITE on mixed state

Still need purification to work well on mixed state

DB-QITE on mixed state

Pure

Mixed

Initial state lies on Bloch sphere surface

Initial state lies inside Bloch sphere

DBAC brings state inside dotted line

(F=0.9)

DBAC brings state closer to \(|0\rangle\) but not inside dotted line (ie more diagonal but still not pure)

DBAC vs HBAC on mixed state

NMR experiments: 3 qubits, 6 steps

See purple and dark purple lines:

Pure and mixed state with 1 step

DBAC

Discussion

Energy reduction

Different 'period'?

|0\rangle

Energy reduction

Possible causes:

  • Depolarising \(\to\) No, Kay has checked
  • Wrong RX gates between experiment and simulation \(\to\) No, because DME (that uses RX) works

Behaviour is consistent across multiple qubit sets

Unused slides

Hardware

  1. Show new Hardware
  2. siZZle Gate
  3. siZZle Compilation for DME: Make DME more natural
  4. First DME with \(n \neq 2\)

Algorithms

  1. A working example of QDP
  2. DB-QITE — New Reset Method using DME
  3. DB-QITE fill a gap left by HBAC

Topics currently covered

Overview of QDP (Quantum Dynamic Programming)

  • Usual quantum computing is static: To change operation, we have to change the circuit
  • Dynamic quantum computing: To change operation, only need to change instruction qubit

Normal way we do quantum computing: Static

Dynamic Quantum Computing

Kjaergaard et al., arxiv:2001.08838

Made with Slides.com