DBAC: Double-bracket Algorithmic Cooling
- About me
- Double-bracket Quantum Imaginary Time Evolution (QITE)
- DBAC Compilation for single qubit
- Quantum Dynamic Programming
Kay Giang - NTU
Master and Bachelor at University of Oxford
- Thesis was in Astrophysics
- Coursework in Quantum Information
Research Assiant at Nanyang Technological University in Singapore
- Multidisciplinary projects - theory, numeric and experimental
- DB-RESET: Algorithmic cooling using double-bracket
About me
Khanh Uyen (Kay) Giang
Outside physics, I enjoy watercolor art, pottery and ice-skating
Double-bracket Quantum Imaginary Time Evolution
(DB-QITE)
Gluza et al., arxiv:2412.04554
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 \(\hat H\)
DB-QITE formula
|\Psi(\tau)\rangle = \frac{e^{-\tau \hat{H}}}{\|e^{-\tau \hat{H}}|\Psi(0)\rangle\|} |\Psi(0)\rangle
Gluza et al. (2412.04554) shows that QITE satisfy:
\begin{align*}
\partial_\tau {|\Psi(\tau) \rangle} = [\Psi(\tau),\hat{H}] |\Psi(\tau) \rangle
\end{align*}
Double-bracket
\frac{\partial \Psi(\tau)}{\partial \tau} = \big[ [\Psi(\tau),\hat H],\Psi(\tau)\big]
In terms of the density matrix \(\Psi(\tau)\):
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
\ket{\psi_{k+1}} = e^{t_k[\psi_k,\hat H]} \ket{\psi_k}
\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]} =
e^{i\sqrt{t}\hat H}e^{i\sqrt{t}\psi}
e^{-i\sqrt{t}\hat H}
e^{-i\sqrt{t}\psi} + \mathcal O(t^{3/2})
Using the group commutator relation:
DB-QITE recursion formula:
Fidelity increase guarantee
\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}}
F_{k+1} \geq 1- (1- t F_0 \Delta)^{k}
Fidelity with ground state after step \(k+1\):
\(F_0\): fidelity between the initial state with ground state
\(\Delta\): spectral gap
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 \(e^{i\sqrt{t_k}\psi_k}\)
DBAC compilation for single qubit case
Compilation for resetting one qubit
\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}}
\(e^{i\sqrt{t_k}\psi_k}\): Density matrix exponentiation (DME)
Single qubit: \(\hat H = \hat Z\)
: \(\delta\)SWAP gate, applying \(e^{-i t \text{SWAP}}\). Compiled using Heisenberg interaction: \(e^{it(XX+YY+ZZ)}\)
DME Circuit
\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}}
\begin{align*}
\hat{E}_t^{(\rho)}(\sigma) &= \text{Tr}_{1}\left[e^{-i t \text{SWAP}}(\rho\otimes\sigma) e^{i t \text{SWAP}}\right]\\
&= \sigma - it[\rho,\sigma] +O(t^2)\\
&= e^{-it\rho} \sigma e^{it\rho} +O(t^2)
\end{align*}
Density matrix exponentiation
The DME channel is:
Repeating \(M\) iterations yields:
\left(\hat{E}_{(t/M)}^{(\rho)}(\sigma)\right)^{M} = e^{-it\rho} \sigma e^{it\rho} + O(t^2/M)
DME Circuit
- This DME protocol was introduced in Kjaergaard et al., arxiv:2001.08838
- In their paper, they use the 3 CNOT decompositions for the \(\delta\text{SWAP}\) gate, and performed it on only 2 qubits
- No literature has performed DME on more qubits
DME Circuit
\(\delta\)SWAP compiled using Heisenberg interaction: \(e^{it(XX+YY+ZZ)}\)
Reason for using ZZ interaction: The entangling operation in transmon qubit is Stark-induced ZZ by level excursions (siZZle)
Example DBAC Circuits
\(k=1\) recursion step (with optimized \(t_k\)):
\(k=2\) recursion steps:
\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}}
DBAC Performance
\(k=2\) recursion steps (3 copies of the reset qubit):
Qubit chip layout
Comparing DBAC with existing protocols
HBAC
- Work iteratively
- NMR/ solid state that have a thermal bath
- Work on any state
- Cool an ensemble of qubit
- Complex unitary decomposition
- Using a full SWAP operation
DBAC
- Work iteratively
- Superconducting system, does not use a bath
- Only work on pure states
- Cool one qubit
- Simple decomposition
- Partial SWAP using hardware-natural decomposition \(\to\) faster
Heat bath algorithmic cooling
Microwave drive
- High fidelity (99.5-99.8%)
- Duration: 500ns
- Hardware specific
- Lower fidelity
- Duration: ~200ns for 1 step
- General purpose
DBAC
Why DBAC?
- Enable qubit chip testing with clear interpretable results and exponential complexity
- Dynamic nature, allows repetition on varied qubit layout
- Proof of principle for a new class of algorithm: dynamic quantum algorithm
Quantum Dynamic Programming (QDP)
A framework that uses copies of the recursive state to implement the recursion step unitary
Static vs Dynamic
- 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
Dynamic Quantum Algorithm
- Question 1: Can quantum information be the source code?
- Answer: Kimmel et al. (arxiv 1608.00281) shows that this model is basis of a universal model for quantum computation
Dynamic Quantum Algorithm
- Questions 2: What would QDP be good for?
-
Example:
- DB-RESET
- Provide a universal circuit that compute the Schmidt spectrum
- For well-behaved recursions, QDP yields accurate results with polynomial depth
Quantum Dynamic Programming
- QDP speed up recursion of the form (single memory call):
\[ U^{(\mathcal{N},\rho)} = V_2e^{i\mathcal{N}(\rho)}V_1\] where \(\mathcal{N}\) is any Hermitian-preserving map - Memory call: Idealized transformation we want to make. It asks for memory (instruction state \(\rho\)).
- General case:
- Problem: We can't implement this naturally in qunatum mechanics
Quantum Dynamic Programming
- Solution: QDP approximate this memory call unitary:
- QDP does this by using memory usage query:
where \(N\) is the operator of the memory usage query, the partial transpose of the Choi matrix corresponding to \(\mathcal{N}\)- Consume (trace out) an instruction state
- Repeat this procedure M times, we obtain
\mathcal{E}_s^{(N,\rho)}(\sigma) = Tr_\rho [e^{-iNs} (\rho\otimes\sigma) e^{iNs}]
E^{(\mathcal{N},\rho)}(\sigma) = e^{i\mathcal{N}(\rho)} \sigma e^{-i\mathcal{N}(\rho)}
\mathcal{E}^{(\mathcal{N},\rho,M)}_{QDP} \coloneqq (\mathcal{E}^{(\mathcal{N},\rho)}_{1/M})^M = E^{(\mathcal{N},\rho)} + O(1/M)
Quantum Dynamic Programming
Example: Density Matrix Exponentiation
- Memory call unitary:
\[E^{(\mathcal{N},\rho)}(\sigma) = e^{i\rho\theta} \sigma e^{-i\rho\theta}\] - Memory call operator: \[\mathcal{N}\rho = \rho \to \mathcal{N} = id\]
- Memory usage query operator: \[N = \delta SWAP\]
- Memory usage query: \[\sigma\to\text{Tr}_\rho[e^{-i\text{SWAP}\delta}(\sigma\otimes\rho) e^{i\text{SWAP}\delta}]\]
Quantum Dynamic Programming
Example: Oblivious Schmidt decomposition
- Bipartite pure state \(|\psi\rangle\) has a Schmidt decomposition:
|\psi\rangle_{AB} = \sum_j \sqrt{\lambda_j} |\phi_j\rangle_A|\chi_j\rangle_B
- Double bracket iteration of OSD:
e^{i\mathcal{N}(\psi_n)} = e^{s_n[D,\rho_n^{(A)}]}\otimes \mathbb{I}_B
- Memory usage query:
- Memory usage query operator:
Summary
- DB-QITE: formulate a recursion relation that implement imaginary time evolution
- DBAC: synthesizes a circuit that reset 1 qubit iteratively
- Quantum dynamic programming (QDP): uses copies of the instruction state to implement the operation
Thank you for listening!
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By Khanh Uyen Giang
