Double-bracket flow quantum algorithm for diagonalization
Marek Gluza
NTU Singapore
slides.com/marekgluza
New quantum algorithm for diagonalization
\hat H \mapsto \hat \mathcal H_\ell = \hat \mathcal U_\ell^\dagger \hat H \hat\mathcal U_\ell
\partial_\ell \hat \mathcal H_\ell = [[A(\hat \mathcal H_\ell),B(\hat \mathcal H_\ell)], \hat \mathcal H_\ell]
no qubit overheads
no controlled-unitaries
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Simple
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Easy
Doesn't spark joy :(
New quantum algorithm for diagonalization
\hat H \mapsto \hat \mathcal H_\ell = \hat \mathcal U_\ell^\dagger \hat H \hat\mathcal U_\ell
\partial_\ell \hat \mathcal H_\ell = [[A(\hat \mathcal H_\ell),B(\hat \mathcal H_\ell)], \hat \mathcal H_\ell]
building useful quantum algorithms
new approach to preparing useful states
building useful variational circuits
tons of fun maths in the appendix
no qubit overheads
no controlled-unitaries
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\partial_\ell \hat H_\ell = [\hat W_\ell, \hat H_\ell]
\hat W_\ell = [\Delta(\hat H_\ell),\sigma(\hat H_\ell)]
\Delta(\hat H_\ell)
\sigma(\hat H_\ell)
Głazek-Wilson-Wegner flow
Restriction to off-diagonal
Restriction to diagonal
\partial_\ell \|\sigma(\hat H_\ell)\|_{\text{HS}}^2 = -2\|\hat W_\ell\|_{\text{HS}}^2 \le 0
as a quantum algorithm
\text{Unitary } \hat U_\ell \text{ such that }
\hat W_\ell = [\Delta(\hat H_\ell),\sigma(\hat H_\ell)]
Głazek-Wilson-Wegner flow
\partial_\ell \|\sigma(\hat H_\ell)\|_{\text{HS}}^2 = -2\|\hat W_\ell\|_{\text{HS}}^2 \le 0
as a quantum algorithm
\hat H_\ell = \hat U_\ell \hat H_0 \hat U_\ell^\dagger
\partial_\ell \hat H_\ell = [\hat W_\ell, \hat H_\ell]
New quantum algorithm for diagonalization
\hat H \mapsto \hat \mathcal H_\ell = \hat \mathcal U_\ell^\dagger \hat H \hat\mathcal U_\ell
\partial_\ell \hat \mathcal H_\ell = [[A(\hat \mathcal H_\ell),B(\hat \mathcal H_\ell)], \hat \mathcal H_\ell]
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\hat D_s(\hat J) = \prod_{\mu\in\{0,1\}^{\times L}} \hat Z_\mu e^{-i s \hat J/D} \hat Z_\mu
\hat C_s(\hat J) = \left( \hat D_{\sqrt{s/K}}(\hat J)^\dagger \;e^{i\sqrt{s/K}\hat J}\;\hat D_{\sqrt{s/K}}(\hat J)\;e^{-i\sqrt{s/K}\hat J}\right)^K
1) Dephasing
2) Group commutator
3) Frame shifting
\partial_\ell \hat \mathcal H_\ell = [[A(\hat \mathcal H_\ell),B(\hat \mathcal H_\ell)], \hat \mathcal H_\ell]
\hat V_k = \hat C_s(\hat J_{1})\; \hat C_s(\hat J_{2})\ldots \hat C_s(\hat J_{k-1}) \approx e^{-s \hat W_1}\;e^{-s \hat W_2}\ldots e^{-s \hat W_{k-1}}
How to understand the continuous flow?
\hat H_1 = e^{s \hat W_1} \hat H e^{-s \hat W_1}
\hat W_1 = [\Delta(\hat H),\sigma(\hat H)]
\hat W_k = [\Delta(\hat H_{k-1}),\sigma(\hat H_{k-1})]
\hat H_k = e^{s \hat W_k} \hat H_{k-1} e^{-s \hat W_k}
\hat H_\text{TFIM} = \sum_{i=1}^{L-1}\hat X_i\hat X_{i+1}+\sum_{i=1}^L \hat Z_i
Piece-wise constant discretization
antihermitian
unitary
\left(i\hat H\right)^\dagger = -i \hat H
How to quantum?
Group commutator
\hat H \mapsto \hat \mathcal H_\ell = \hat \mathcal U_\ell^\dagger \hat H \hat\mathcal U_\ell
\partial_\ell \hat \mathcal H_\ell = [[A(\hat \mathcal H_\ell),B(\hat \mathcal H_\ell)], \hat \mathcal H_\ell]
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Want
\hat H_1 = e^{s \hat W_1} \hat H e^{-s \hat W_1}
\hat W_1 = [\Delta(\hat H),\sigma(\hat H)]
e^{is\Delta(\hat H)}e^{is\hat H}e^{-is\Delta(\hat H)}e^{-is\hat H}= e^{-s^2[\Delta(\hat H),\sigma(\hat H)]} +\hat E^\text{(GC)}
New bound
\|\hat E^\text{(GC)}\| \le 4\|\hat H\|\,\|[\Delta(\hat H),\sigma(\hat H)]\| s^3
Group commutator
\hat H \mapsto \hat \mathcal H_\ell = \hat \mathcal U_\ell^\dagger \hat H \hat\mathcal U_\ell
\partial_\ell \hat \mathcal H_\ell = [[A(\hat \mathcal H_\ell),B(\hat \mathcal H_\ell)], \hat \mathcal H_\ell]
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Want
\hat H_1 = e^{s \hat W_1} \hat H e^{-s \hat W_1}
\hat W_1 = [\Delta(\hat H),\sigma(\hat H)]
e^{is\Delta(\hat H)}e^{is\hat H}e^{-is\Delta(\hat H)}e^{-is\hat H}= e^{- s^2[\Delta(\hat H),\sigma(\hat H)]} +\hat E^\text{(GC)}
How to get ?
\hat D_s(\hat H) \approx e^{is\Delta(\hat H)}
\hat C_s(\hat J) = \hat D_{\sqrt{s}}(\hat J)^\dagger \; e^{i\sqrt{s}\hat J}\; \hat D_{\sqrt{s}}(\hat J)\; e^{-i\sqrt{s}\hat J}
\approx e^{-s \hat W(\hat J)}
1,
1,
1,
1,
0,
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Phase flip unitaries
\mu = (
)
N
S
N
S
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S
\hat Z
\mathbb 1
\mathbb 1
\otimes
\otimes
\otimes
\otimes
\otimes
\otimes
\otimes
\hat Z
\hat Z
\hat Z
\mathbb 1
\mathbb 1
0,
1,
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1,
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1
\mu = (
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1
\otimes
\mathbb 1
\otimes
\mathbb 1
\otimes
\hat Z
\otimes
1
\otimes
\mathbb 1
\otimes
\hat Z
\otimes
\hat Z
N
S
N
S
N
S
Phase flip unitaries
Evolution under dephased generators
\hat H \mapsto \hat H_\ell = \hat U_\ell^\dagger \hat H \hat U_\ell
\partial_\ell \hat H_\ell = [[A(\hat H_\ell),B(\hat H_\ell)], \hat H_\ell]
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\hat Z_\mu e^{-i s \hat J} \hat Z_\mu = e^{-is \hat Z_\mu \hat J\hat Z_\mu}
\hat D_s(\hat J) \approx \prod_{\mu\in R} \hat Z_\mu e^{-i s \hat J/D} \hat Z_\mu
We can make it efficient:
\Delta(\hat J)= \frac 1 {\dim (\hat J)} \sum_{\mu\in\{0,1\}^{\times L}} \hat Z_\mu \hat J \hat Z_\mu
R \subseteq \{0,1\}^{\times L}
Use unitarity
and repeat many times
New quantum algorithm for diagonalization
\hat H \mapsto \hat \mathcal H_\ell = \hat \mathcal U_\ell^\dagger \hat H \hat\mathcal U_\ell
\partial_\ell \hat \mathcal H_\ell = [[A(\hat \mathcal H_\ell),B(\hat \mathcal H_\ell)], \hat \mathcal H_\ell]
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\hat D_s(\hat J) = \prod_{\mu\in\{0,1\}^{\times L}} \hat Z_\mu e^{-i s \hat J/D} \hat Z_\mu
\hat C_s(\hat J) = \left( \hat D_{\sqrt{2s/K}}(\hat J)^\dagger \;e^{i\sqrt{2s/K}\hat J}\;\hat D_{\sqrt{2s/K}}(\hat J)\;e^{-i\sqrt{2s/K}\hat J}\right)^K
\hat V_k = \hat V_{k-1}\hat C_s(\hat J_{k-1})
\hat J_{k-1} = \hat V_{k-1}^\dagger \hat H \hat V_{k-1}
1) Dephasing
2) Group commutator
3) Frame shifting
\partial_\ell \hat \mathcal H_\ell = [[A(\hat \mathcal H_\ell),B(\hat \mathcal H_\ell)], \hat \mathcal H_\ell]
\hat V_k = \hat C_s(\hat J_{1})\; \hat C_s(\hat J_{2})\ldots \hat C_s(\hat J_{k-1}) \approx e^{-s \hat W_1}\;e^{-s \hat W_2}\ldots e^{-s \hat W_{k-1}}
What's going on?
Double-bracket flow
Unitary
Satisfies a generalization of the Heisenberg equation
\hat H_\ell = \hat U_\ell^\dagger \hat H \hat U_\ell
\partial_\ell \hat H_\ell = [[A(\hat H_\ell),B(\hat H_\ell)], \hat H_\ell]
\hat U_\ell
GWW is a particular example
\partial_\ell \hat H_\ell = [[\Delta(\hat H_\ell),\sigma(\hat H_\ell)], \hat H_\ell]
transformation of
\hat H
\hat H_s = e^{s \hat W^{(Z)}} \hat H e^{-s \hat W^{(Z)}}
\hat W^{(Z)} = [\Delta^{(Z)}(\hat H), \hat H ],
\partial_s \|\sigma(\hat H_s)\|_{\text{HS}}^2 = -2\langle \hat W^{(Z)},\hat W\rangle_{\text{HS}}
\Delta^{(Z)}(\hat H) \text{ - diagonal}
Variational double-bracket flows
that are diagonalizing
\partial_\ell \|\sigma(\hat H_\ell)\|_{\text{HS}}^2 = -2\|\hat W_\ell\|_{\text{HS}}^2 \le 0
How well does it work?
Variational flow example
Notice the steady increase of diagonal dominance.
Variational vs. GWW flow
Notice that degeneracies limit GWW diagonalization but variational brackets can lift them.
GWW for 9 qubits
Notice the spectrum is almost converged.
How does it work again?
GWW for 9 qubits
Notice that some of them are essentially eigenstates!
What else is there?
Linear programming
Matching optimization
Diagonalization
Sorting
QR decomposition
Toda flow
\partial_\ell \hat H_\ell = [[A(\hat H_\ell), B(\hat H_\ell)] , \hat H_\ell]
Double-bracket flow
Runtime-boosting heuristics
Analytical convergence analysis
Group commutator bound
Hasting's conjecture
Relation to other quantum algorithms
Code is available on Github
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Double-bracket flow quantum algorithm for diagonalization
Material science?
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