New quantum algorithms based on Riemannian optimization
Marek Gluza
NTU Singapore
Scan QR code or go to slides.com/marekgluza
Watch on youtube: https://www.youtube.com/watch?v=PLVkuqPemVs
Marek Gluza
The double-bracket quantum algorithms roadmap
I grew up around these mountains where Poland meets Czech Republic and Slovakia (in Europe)
June '22: Single-author double-bracket proposal
This talk is an overview of 4 years of my research on double-bracket quantum algorithms:
- Never worked on quantum algorithms before \(\rightarrow\) 9 papers, 1 experiment
- Never participated in a research program \(\rightarrow\) lead a collaboration of 25 co-authors
- It was the mathematical observations guiding us \(\rightarrow\) Riemannian geometry is key!
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[9]
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[3,4]
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Click these links at slides.com/marekgluza
October '21: Arrived to Singapore
Marek Gluza
The double-bracket quantum algorithms roadmap
...the outline of the talk
Today, I will tell you about double-bracket quantum algorithms
Part 1:
- What are quantum algorithms?
- What is quantum computing?
Part 3:
- What can they be used for?
- What are good tasks for quantum computers?
Part 2:
- Riemannian geometry of the unitary group
Part 4: Just basic properties of the unitary group facilitate quantum algorithm design
What is a quantum algorithm?
0
0
0
0
C
What is a quantum computer?
It's an experimental setup A which includes a quantum system B and that setup A allows to manipulate the quantum state of B. This is modeled by mulitplying unitary matrices to vectors from a finite-dimensional vector space.
(This monumental structure is the insides of the dilution refrigerator shielding qubits from noise)
A - basket
B - fruits
Cat - noise
A - quantum computer
B - qubits
C - noise
What is a universal quantum computer?
|\psi\rangle = a | 0 \rangle + b | 1\rangle
|\psi\rangle = a | 00 \rangle + b | 10\rangle+c | 01 \rangle + b | 11\rangle
|\psi\rangle = a | 000 \rangle + b | 100\rangle+c | 010 \rangle + d | 100\rangle+
e | 011 \rangle + f | 101\rangle+g | 110 \rangle +h | 111\rangle
|\psi\rangle = a | 0000 \rangle + b | 1000\rangle+c | 0100 \rangle + d | 1000\rangle+
e | 0011 \rangle + f | 0101\rangle+g | 1010 \rangle +h | 1100\rangle + \ldots
1 qubit
2 qubits
3 qubits
4 qubits
|\psi\rangle = U |00\ldots0\rangle \Rightarrow U \approx U_c
it can approximate any state preparation via circuits
It is a machine for realizing such linear combinations in nature
0
0
0
0
C
Operating a quantum computer is all about the group of unitary matrices
M,M' \in U(d) \Rightarrow (M M')(MM')^\dagger = M (M'M'^\dagger) M^\dagger
= M M^\dagger = I
\Rightarrow (M M')\in U(d)
\ket{\psi'} = M \ket\psi \Rightarrow \ket{\psi''} = M'\ket{\psi'} = M'M \ket\psi
Think of rotations on a sphere
U(d) = \{ M\in \mathbb C^{d\times d}: M^{-1} = M^\dagger\}
|\psi'\rangle
|\psi\rangle
|\psi''\rangle
|\psi\rangle = U |00\ldots0\rangle
Operating a quantum computer is all about the group of unitary matrices
\exist G_1,G_2,\ldots G_K \in U(d) \mathrm{~such~that}
Think of rotations on a sphere
Universal quantum computation means: Composing sufficiently many basic unitary matrices (elementary quantum gates \(G_1,G_2,\ldots, G_K\)) can approximate any other 'larger' unitary.
(finding these sequences = task of unitary synthesis)
U(d) = \{ M\in \mathbb C^{d\times d}: M^{-1} = M^\dagger\}
\forall M \in U(d)~\exist \mathrm{sequence~s.t.} ~M\approx G_1 G_5G_6G_1 G_K G_1 G_2\ldots
4 stages of creating quantum algorithms
Guidelines for using quantum computing
Stage 1: Think. What is the goal?
Important problems that are difficult yet doable.
Encode what we know in \(\vec{v}_{input}\).
Decode information from \(\vec{v}_{output}\).
Find effective heuristics to reduce the runtime of rotations \(R_k\)
Find rotations such that
\(\vec{v}_{output} \approx R_1 R_2 \ldots R_n \vec{v}_{input}\)
Stage 2: Design. How to encode task in quantum mechanics?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
Stage 4: Run it.
What instructions to send?
Stage 3: Algorithm. How to find \(\vec{v}_{output}\)?
Guidelines for extracting utility from quantum computing
Stage 1: Think. What is the goal?
- Bridge between needs of humans and technological feasibility.
- Exchange domain expertise with industry partners.
- My work: General-purpose optimization solver in quantum computing based on non-Euclidean geometry.
- My expertise: Algorithms optimized for execution on leading prototypes.
Stage 4: Run it.
What instructions to send?
Stage 2: Design. How to encode task in quantum mechanics?
Stage 3: Algorithm. How to find \(\vec{v}_{output}\)?
Marek Gluza
The double-bracket quantum algorithms roadmap
...the outline of the talk
Today, I will tell you about double-bracket quantum algorithms
Part 1:
- What are quantum algorithms?
- What is quantum computing?
Part 3:
- What can they be used for?
- What are good tasks for quantum computers?
Part 2:
- Riemannian geometry of the unitary group
Part 4: Just basic properties of the unitary group facilitate quantum algorithm design
Mathematically, states of the quantum computer are like arrows pointing from the center of the sphere to its surface.
Observation leading to double-bracket quantum algorithms: Earth is not flat. I.e., when we walk along of the equator, we think we are going straight but eventually we will wrap around it.
Fixing a direction and rotating the arrow, corresponds to a type of of quantum computing operation.
Stage 3: Algorithm. How to find \(\vec{v}_{output}\)?
Riemannian geometry underlying quantum algorithms
On a flat surface DOWN-LEFT-UP-RIGHT will return to point of origin.
Stage 3: Algorithm. How to find \(\vec{v}_{output}\)?
Riemannian geometry underlying quantum algorithms
On a flat surface DOWN-LEFT-UP-RIGHT will return to point of origin.
On a curved surface SOUTH-WEST-NORTH-EAST will spiral way.
South
West
East
North
Stage 3: Algorithm. How to find \(\vec{v}_{output}\)?
Riemannian geometry underlying quantum algorithms
South
West
East
North
My work on double-bracket quantum algorithms shows how to use this spiraling effect to implement non-Euclidean gradient descent in quantum computing.
Regular machine learning fails for quantum computing but our generalization works. The 'failed' machine learning is still key for us - as a warm-start!
(Physical Review Letters '26)
Stage 3: Algorithm. How to find \(\vec{v}_{output}\)?
Riemannian geometry underlying quantum algorithms
Riemannian geometry is essential for quantum computation
- The unitary group \(U(d)\) is a Riemannian manifold
- It is an embedded manifold \(U(d) = \{M\in \mathbb C^{d\times d}:~M^{-1}=M^\dagger\}\)
- The tangent space is \(\{W\in \mathbb C^{d\times d}:~W^{\dagger}= -W\} \simeq\{iH \mathrm{~where~} H=H^\dagger\in \mathbb C^{d\times d}\} \)
- The geodesics are matrix exponentials \( \{e^{sW}\}_{s\in\mathbb R} \subset U(d)\) or \( \{e^{isH}\}_{s\in\mathbb R} \subset U(d)\)
- Computing a "gradient" must output an element of the tangent space
\mathbb C^{d\times d}
U(d)
g
W
iH
g^\dagger = -g
e^{s g}
g
g^\dagger = -g
\(\partial_{i,j}\) points to the interior, not tangential
Keep this in mind for later: Unlike in flat space, these 4 steps spiral away from the point of origin
Riemannian geometry is essential for quantum computation
- The unitary group \(U(d)\) is a Riemannian manifold
- The tangent space is \(\{H\in \mathbb C^{d\times d}:~H^{\dagger}= H\} \)
- The geodesics are matrix exponentials \( \{e^{isH}\}_{s\in\mathbb R} \subset U(d)\)
- \(U(d)\) is a curved manifold
e^{is A}e^{is B}e^{-is A}e^{-is B} \neq 1
e^{is A}
e^{isB}
e^{-isB}
e^{-is A}
Operating a quantum computer is all about the group of unitary matrices
\forall A=A^\dagger, B=B^\dagger: ~~M = e^{ [A,B]}\in U(d)
Think of rotations on a sphere
Fact 3: The Lie bracket of two 'velocities' is again a velocity
U(d) = \{ M\in \mathbb C^{d\times d}: M^{-1} = M^\dagger\}
e^{isA}e^{isB}e^{-isA}e^{-isB} = e^{-s^2[A,B]}+O(s^3)
e^{is A}
e^{isB}
e^{-isB}
e^{-is A}
Marek Gluza
Marek Gluza
The double-bracket quantum algorithms roadmap
...the outline of the talk
Today, I will tell you about double-bracket quantum algorithms
Part 1:
- What are quantum algorithms?
- What is quantum computing?
Part 3:
- What can they be used for?
- What are good tasks for quantum computers?
Part 2:
- Riemannian geometry of the unitary group
Part 4: Just basic properties of the unitary group facilitate quantum algorithm design
4 stages of creating quantum algorithms
1. Design choice:
How to go about it?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
3. Circuit compilation:
What gates to do?
0. Problem choice:
What challenge to take up?
2. Unitary synthesis:
How to do it?
Double-bracket quantum algorithms:
Systematic framework for unitary synthesis
4 stages of creating quantum algorithms
Quantum computing cannot be useful if
- The problem can be solved quickly on classical computers
- Solving the problem isn't important
- Solving the problem is too hard for the quantum computer
Classical computation
Key question in current quantum computation:
Find an important problem which is difficult yet doable
Millenials
What challenge to take up?
Quantum computing cannot be useful if
- The problem can be solved quickly on classical computers
- Solving the problem isn't important
- Solving the problem is too hard for the quantum computer
Key question in current quantum computation:
Find an important problem which is difficult yet doable
What challenge to take up?
Materials science?
Why materials science? Imagine improving photovoltaics by 1%.
What needs to be done? Perform \(\bra\psi H\ket\psi\) optimization.
How? Quantum circuit synthesis.
4 stages of creating quantum algorithms
1. Design choice:
How to go about it?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
3. Circuit compilation:
What gates to do?
0. Problem choice:
What challenge to take up?
2. Unitary synthesis:
How to do it?
Double-bracket quantum algorithms:
Systematic framework for unitary synthesis
4 stages of creating quantum algorithms
E(\psi) = \langle \psi| H | \psi\rangle
4 stages of creating quantum algorithms
1. Design choice:
How to go about it?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
3. Circuit compilation:
What gates to do?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
0. Problem choice:
What challenge to take up?
2. Unitary synthesis:
How to do it?
4 stages of creating quantum algorithms
Imaginary-time evolution
\ket\psi\rightarrow |\psi(\tau)\rangle
H = \sum_{k=1}^{D} \lambda_k \ket{\lambda_k}\bra{\lambda_k}
e^{-\tau H}|\psi\rangle = \langle\lambda_0|\psi\rangle \ket{\lambda_0} +\sum_{k=1}^{D} e^{-\tau\lambda_k} \langle\lambda_k|\psi\rangle \ket{\lambda_k}
|\psi\rangle = \langle\lambda_0|\psi\rangle \ket{\lambda_0} +\sum_{k=1}^{D} \langle\lambda_k|\psi\rangle \ket{\lambda_k}
\lim_{\tau\rightarrow \infty}\ket{\psi(\tau)} = \ket{\lambda_0}
4 stages of creating quantum algorithms
1. Design choice:
How to go about it?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
3. Circuit compilation:
What gates to do?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
0. Problem choice:
What challenge to take up?
2. Unitary synthesis:
How to do it?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
= e^{\tau[|\psi\rangle\langle\psi|,H]}|\psi\rangle +O(\tau^2)
4 stages of creating quantum algorithms
Marek Gluza
The double-bracket quantum algorithms roadmap
...the outline of the talk
Today, I will tell you about double-bracket quantum algorithms
Part 1:
- What are quantum algorithms?
- What is quantum computing?
Part 3:
- What can they be used for?
- What are good tasks for quantum computers?
Part 2:
- Riemannian geometry of the unitary group
Part 4: Just basic properties of the unitary group facilitate quantum algorithm design
4 stages of creating quantum algorithms
1. Design choice:
How to go about it?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
3. Circuit compilation:
What gates to do?
0. Problem choice:
What challenge to take up?
2. Unitary synthesis:
How to do it?
Double-bracket quantum algorithms:
Systematic framework for unitary synthesis
4 stages of creating quantum algorithms
E(\psi) = \langle \psi| H | \psi\rangle
Operating a quantum computer is all about the group of unitary matrices
\forall A=A^\dagger, B=B^\dagger: ~~M = e^{ [A,B]}\in U(d)
Think of rotations on a sphere
The Lie bracket of two 'velocities' is again a velocity:
Check: \([A,B]^\dagger = (AB- BA)^\dagger = B^\dagger A^\dagger -A^\dagger B^\dagger = -[A,B]\)
M^\dagger = (e^{ [A,B]})^\dagger = e^{ [A,B]^\dagger}=e^{ -[A,B]}=M^{-1}
U(d)
e^{ [A,B]}
\([A,B]^\dagger = -[A,B]\)
Check: \([A,B]^\dagger = -[A,B]\)
U(d) = \{ M\in \mathbb C^{d\times d}: M^{-1} = M^\dagger\}
Operating a quantum computer is all about the group of unitary matrices
\forall A=A^\dagger, B=B^\dagger: ~~M = e^{ [A,B]}\in U(d)
Think of rotations on a sphere
Fact 3: The Lie bracket of two 'velocities' is again a velocity
U(d) = \{ M\in \mathbb C^{d\times d}: M^{-1} = M^\dagger\}
e^{isA}e^{isB}e^{-isA}e^{-isB} = e^{-s^2[A,B]}+O(s^3)
e^{is A}
e^{isB}
e^{-isB}
e^{-is A}
Operating a quantum computer is all about the group of unitary matrices
U(d) = \{ M\in \mathbb C^{d\times d}: M^{-1} = M^\dagger\}
Think of rotations on a sphere
Fact 3: The Lie bracket of two 'velocities' is again a velocity
My quantum algorithms use such unitary matrices - they are implementing Riemannian gradient descent
e^{isA}e^{isB}e^{-isA}e^{-isB} = e^{-s^2[A,B]}+O(s^3)
\forall A=A^\dagger, B=B^\dagger: ~~M = e^{ [A,B]}\in U(d)
The main tool of double-bracket quantum algorithms
e^{is A}
e^{isB}
e^{-isB}
e^{-is A}
E(\psi) = \langle \psi| H | \psi\rangle
| \psi\rangle \mapsto \ket{\psi'}= e^{is A}|\psi\rangle
Note that choosing \(A=H\) doesn't change the energy
E(\psi') = \langle \psi| e^{-is H} He^{isH} | \psi\rangle = E(\psi)
Let's find which directions \(A\) are more useful!
\(\partial_{i,j}\) points to the interior, not tangential so which direction is best?
g
W
iH
iA
Text
We need to find a tangential direction which lowers the energy of \(\ket\psi\)
E(\psi) = \langle \psi| H | \psi\rangle
| \psi\rangle \mapsto \ket{\psi'}= e^{is A}|\psi\rangle
Let's see how a direction \(A=A^\dagger\) changes the energy of \(\ket\psi\)
Tr(ABC) = Tr(CAB)\\
\Downarrow \\
Tr(|\psi\rangle\langle \psi| [H,A] ) = Tr([|\psi\rangle\langle \psi|,H]A)
\partial_s \langle \psi| e^{-isA}He^{isA}| \psi\rangle_{|s=0} = -i\langle \psi| [A,H]| \psi\rangle
=- iTr\left[|\psi\rangle\langle\psi| [A,H]\right]
This bracket is called the Riemannian gradient
g
W
iH
iA
g \equiv -i[|\psi\rangle\langle \psi|,H]
|\psi'\rangle = e^{s[|\psi\rangle\langle \psi|,H]} |\psi\rangle
|\psi\rangle \mapsto e^{isg} |\psi\rangle
|\psi''\rangle = e^{s[|\psi'\rangle\langle \psi'|,H]} |\psi'\rangle
Double-bracket quantum algorithms:
Systematic framework for implementing exponentials of commutators on quantum computers. This uncovered new unitary synthesis formulas.
e^{is A}
e^{isB}
e^{-isB}
e^{-is A}
Double-bracket flows
\partial_t O(t)= i[H,O(t)]
\partial_t \rho(t)= i[H,\rho(t)]
\partial_t A(t)= [A(t),[A(t),B]]
H \sim -i[A(t),B]
|
Heisenberg equation |
Linear, variable: observable |
|---|---|
|
Schroedinger equation |
Linear, variable: density matrix |
|
Double-bracket flow |
Non-linear, variable: density matrix or observable The solution is a unitary rotation because |
Why double a bracket?
\partial_t \hat A(t) = i [ \hat H(t), \hat A(t)]
\partial_t \hat A(t) = [ \hat H(t), [\hat H(t), \hat A(t)]]
=
2 qubit unitary
Canonical
\partial_t \hat A(t) = [ \hat A(t), [\hat A(t), \hat B]]
Double-bracket quantum algorithms
are inspired by double-bracket flows
and allow to perform optimization through short quantum computations
Non-Euclidean geometry leads to quantum algorithms
Next: How to get $$ from a quantum computer?
We need to optimize among all operations of the quantum computer to get a low-energy state
(Non-Euclidean optimization)
E(\psi) = \langle \psi| H | \psi\rangle
\nabla_W f(x) = \langle g, W\rangle
\nabla_A E(\psi) =
\langle- i[|\psi\rangle\langle \psi|,H] , A\rangle_{\rm{HS}}
\\\Downarrow\\
g \equiv -i[|\psi\rangle\langle \psi|,H]
|\psi'\rangle = e^{s[|\psi\rangle\langle \psi|,H]} |\psi\rangle
|\psi\rangle \mapsto e^{isg} |\psi\rangle
|\psi''\rangle = e^{s[|\psi'\rangle\langle \psi'|,H]} |\psi'\rangle
Double-bracket quantum algorithms:
Systematic framework for implementing exponentials of commutators on quantum computers. This uncovered new unitary synthesis formulas.
Riemannian gradient: Unique vector \(g\) in the tangent space such that the directional derivative is a projection onto \(g\)
e^{is A}
e^{isB}
e^{-isB}
e^{-is A}
Marek Gluza
The double-bracket quantum algorithms roadmap
...the outline of the talk
Today, I will tell you about double-bracket quantum algorithms
Part 1:
- What are quantum algorithms?
- What is quantum computing?
Part 3:
- What can they be used for?
- What are good tasks for quantum computers?
Part 2:
- Riemannian geometry of the unitary group
Part 4: Just basic properties of the unitary group facilitate quantum algorithm design
4 stages of creating quantum algorithms
1. Design choice:
How to go about it?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
0. Problem choice:
What challenge to take up?
Double-bracket quantum algorithms:
Systematic framework for unitary synthesis
4 stages of creating quantum algorithms
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
E(\psi) = \langle \psi| H | \psi\rangle
e^{s[|\psi\rangle\langle \psi|,H]} |\psi\rangle
3. Circuit compilation:
What gates to do?
2. Unitary synthesis:
How to do it?
4 stages of creating quantum algorithms
\frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
= e^{\tau[|\psi\rangle\langle\psi|,H]}|\psi\rangle +O(\tau^2)
Product formula approximation:
\(e^{\tau[|\psi\rangle\langle\psi|,H]} = e^{i\sqrt{\tau}H}e^{i\sqrt{\tau}|\psi\rangle\langle\psi|}e^{-i\sqrt{\tau}H}e^{-i\sqrt{\tau}|\psi\rangle\langle\psi|} + O(\tau^{3/2})\)
4 stages of creating quantum algorithms
2. Unitary synthesis:
How to do it?
1. Design choice:
How to go about it?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
3. Circuit compilation:
What gates to do?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
0. Problem choice:
What challenge to take up?
= e^{i\sqrt{\tau}H}e^{i\sqrt{\tau}|\psi\rangle\langle\psi|}e^{i\sqrt{\tau}H}|\psi\rangle + O(\tau^{3/2})
3. Circuit compilation:
What gates to do?
Product formula approximation:
\(e^{\tau[|\psi\rangle\langle\psi|,H]} = e^{i\sqrt{\tau}H}e^{i\sqrt{\tau}|\psi\rangle\langle\psi|}e^{-i\sqrt{\tau}H}e^{-i\sqrt{\tau}|\psi\rangle\langle\psi|} + O(\tau^{3/2})\)
Quantum algorithm DB-QITE - iterate recursively:
- Define \(|{\psi_k}\rangle = U_k |0\rangle\)
- Use \(e^{is |\psi_k\rangle\langle\psi_k|} = U_ke^{is |0\rangle\langle0|}U_k^\dagger \)
- Recursively iterate \( U_{k+1} = e^{is H} U_k e^{is |0\rangle\langle 0|} U_k^\dagger e^{-is H} U_k\)
3 stages of creating quantum algorithms
1. Design choice:
How to go about it?
3. Circuit compilation:
What gates to do?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
0. Problem choice:
What challenge to take up?
e^{itH}\\
e^{it|\psi\rangle\langle\psi|}
4 stages of creating quantum algorithms
2. Unitary synthesis:
How to do it?
(accepted at PRL)
3. Circuit compilation:
What gates to do?
Product formula approximation:
\(e^{\tau[|\psi\rangle\langle\psi|,H]} = e^{i\sqrt{\tau}H}e^{i\sqrt{\tau}|\psi\rangle\langle\psi|}e^{-i\sqrt{\tau}H}e^{-i\sqrt{\tau}|\psi\rangle\langle\psi|} + O(\tau^{3/2})\)
Quantum algorithm DB-QITE - iterate recursively:
- Define \(|{\psi_k}\rangle = U_k |0\rangle\)
- Use \(e^{is |\psi_k\rangle\langle\psi_k|} = U_ke^{is |0\rangle\langle0|}U_k^\dagger \)
- Recursively iterate \( U_{k+1} = e^{is H} U_k e^{is |0\rangle\langle 0|} U_k^\dagger e^{-is H} U_k\)
3 stages of creating quantum algorithms
1. Design choice:
How to go about it?
3. Circuit compilation:
What gates to do?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
0. Problem choice:
What challenge to take up?
e^{itH}\\
e^{it|\psi\rangle\langle\psi|}
4 stages of creating quantum algorithms
2. Unitary synthesis:
How to do it?
(accepted at PRL)
Numerical results for DB-QITE:
DB-QITE:
- Define \(|{\psi_k}\rangle = U_k |0\rangle\)
- Recursively iterate \( U_{k+1} = e^{is H} U_k e^{is |0\rangle\langle 0|} U_k^\dagger e^{-is H} U_k\)
Then:
Quantinuum
(accepted at PRL)
A crisis in quantum algorithm design:
Will quantum computers be fast?
1. Design choice:
How to go about it?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
2. Unitary synthesis:
How to do it?
3. Circuit compilation:
What gates to do?
0. Problem choice:
What challenge to take up?
Adiabatic "QAOA" approach:
Slowly change the Hamiltonian to "quantumly" tunnel into the solution - not fast enough see e.g. arxiv:2510.06337
Journalist questions vs. hype:
- What was done?
- Where are its limitations?
- Why is it faster?
(Pros and cons of heuristics)
1. Design choice:
How to go about it?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
3. Circuit compilation:
What gates to do?
0. Problem choice:
What challenge to take up?
Double-bracket quantum algorithms are guaranteed to converge. They can pick-up after learning of an optimized circuit has gotten stuck.
Brute-force idea:
Select geodesic directions and learn how long step lengths - will get stuck arxiv:1803.11173
See Nature Comm. referee reports of arxiv:1803.11173
A crisis in quantum algorithm design:
Will quantum computers be fast?
S. Carrazza
August '24: Very few!
August '23: Maybe few?
How many quantum gates are needed?
(Pros and cons of heuristics)
4 stages of creating quantum algorithms
1. Design choice:
How to go about it?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
3. Circuit compilation:
What gates to do?
0. Problem choice:
What challenge to take up?
2. Unitary synthesis:
How to do it?
Double-bracket quantum algorithms:
Systematic framework for unitary synthesis
4 stages of creating quantum algorithms
Double-bracket quantum algorithms
Click these links at slides.com/marekgluza
| Diagonalization | https://arxiv.org/abs/2206.11772 | ||
|---|---|---|---|
| Imaginary-time evolution | https://arxiv.org/abs/2412.04554 | ||
| Quantum signal processing | https://arxiv.org/abs/2504.01077 | ||
| Grover's search | https://arxiv.org/abs/2507.15065 | Approximates ITE |
U^\dagger H U =
\begin{pmatrix}
\lambda_0&&\\
&\lambda_1&\\
&&\ddots
\end{pmatrix}
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
\frac{ \prod_{k=1}^K(H-z_k I)|\psi\rangle}{\| \prod_{k=1}^K(H-z_k I)|\psi\rangle\|}
\prod_{k=1}^Ke^{i\theta_k|\psi_k\rangle\langle\psi_k|} e^{s_k[|\psi_k\rangle\langle\psi_k|,H]}|\psi\rangle
H' = e^{-s[D,H]}H e^{s[D,H]}
e^{s[|\psi\rangle\langle\psi|,H]}|\psi\rangle +O(\tau^2)
(1-2|\psi\rangle\langle\psi|)\times e^{i t H_f}
e^{s[|\psi\rangle\langle\psi|,H]}|\psi\rangle
Solve the unitary synthesis problem in all these cases through Riemannian gradients!
Double-bracket quantum algorithms
- Coherently implement Riemannian gradient steps
-
Give rigorous unitary synthesis for
- imaginary-time evolution
- quantum signal processing
- diagonalization unitaries
- Grover's as an approximation to imaginary-time evolution
- Training quantum circuits from data doesn't work well, unlike in classical machine learning applications. However those variational learning methods are great for warm-starting double-bracket quantum algorithms!
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N. Ng
Z. Holmes
R. Zander
R. Seidel
Y. Suzuki
B. Tiang
J. Son
S. Carrazza
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What is quantum signal processing?
|\psi'\rangle = e^{-\tau H} |\psi\rangle = \sum_k \langle\lambda_k|\psi\rangle e^{-\tau \lambda_k }|\lambda_k\rangle \rightarrow 0
|\psi'\rangle = \frac{e^{-\tau H} |\psi\rangle}{\|e^{-\tau H} |\psi\rangle\|} \rightarrow |\lambda_0\rangle
|\psi'\rangle = \frac{P(H) |\psi\rangle}{\|P(H) |\psi\rangle\|} \rightarrow |\lambda_0\rangle
P(x)=\prod_{k=1}^K(x-x_k)\approx e^{-\tau x}
|\psi'\rangle = \frac{P(H) |\psi\rangle}{\|P(H) |\psi\rangle\|} \rightarrow \frac{H^{-1}|\psi\rangle}{\|H^{-1} |\psi\rangle\|}
P(x)\approx 1/x
Example 1: Exponential function
Example 2: Polynomial approximation to exponential
Example 3: Polynomial approximation to inversion
P(H)=\prod_{k=1}^K(H-x_k)\approx e^{-\tau H}
Quantum signal processing maps between states according to a polynomial filter
|\psi'\rangle = \frac{P(H) |\psi\rangle}{\|P(H) |\psi\rangle\|}
Approach 2. Exponentials of commutators
- Recursive ~ exponentially long circuits
- Double-bracket quantum algorithms link to Riemannian gradients
Approach 1. Block-encodings
- Probablistic ~ exponentially unlikely postselection
- Grand unification of quantum algorithms https://arxiv.org/abs/2105.02859
|\psi'\rangle = \frac{P(H) |\psi\rangle}{\|P(H) |\psi\rangle\|} = U_\psi(P) |\psi\rangle
U_\psi(P)
Next:
- we will consider \(1-\tau H \approx e^{-\tau H}\)
- we will find \(U_\psi(P = 1-\tau H)\)
- discuss how to generalize to any QSP
|\psi'\rangle = \frac{P(H) |\psi\rangle}{\|P(H) |\psi\rangle\|} = U_\psi(P) |\psi\rangle
U_\psi(P)
|\psi'\rangle = \frac{P(H) |\psi\rangle}{\|P(H) |\psi\rangle\|} = U_\psi(P) |\psi\rangle
U_\psi(P)
|\phi'\rangle = e^{s[|\phi\rangle\langle\phi|,H]} |\phi\rangle
U_\phi(P) = e^{s[|\phi\rangle\langle\phi|,H]}
We will show that
\(P(H) = 1-\tau_sH\)
Ansatz:
|\psi'\rangle = e^{s[|\psi\rangle\langle \psi|,H]} |\psi\rangle
|\psi'\rangle = \sum_{n=0}^\infty \frac{s^n}{n!} ([|\psi\rangle\langle \psi|,H])^n |\psi\rangle
[|\psi\rangle\langle \psi|,H] |\psi\rangle = |\psi\rangle\langle \psi|H |\psi\rangle - H|\psi\rangle\langle \psi |\psi\rangle
[|\psi\rangle\langle \psi|,H] |\psi\rangle = (E-H) |\psi\rangle
Double-bracket ansatz:
\(n=1\):
([|\psi\rangle\langle \psi|,H])^2 |\psi\rangle = [|\psi\rangle\langle \psi|,H](E-H) |\psi\rangle
([|\psi\rangle\langle \psi|,H])^2 |\psi\rangle = E(E-H) |\psi\rangle - [|\psi\rangle\langle \psi|,H] H |\psi\rangle
= - V|\psi\rangle
([|\psi\rangle\langle \psi|,H])^{2k} |\psi\rangle = (- 1)^k V^k|\psi\rangle
([|\psi\rangle\langle \psi|,H])^{2k+1} |\psi\rangle = (- 1)^k V^k(E-H)|\psi\rangle
[|\psi\rangle\langle \psi|,H] |\psi\rangle = (E-H) |\psi\rangle
\(n=1\):
\(n=2\):
Y. Suzuki
= E^2 |\psi\rangle - EH |\psi\rangle - |\psi\rangle\langle \psi|H^2 |\psi\rangle + E H|\psi\rangle
V =\langle \psi |H^2|\psi\rangle - \langle \psi |H|\psi\rangle^2
([|\psi\rangle\langle \psi|,H])^{2k} |\psi\rangle = (- 1)^k V^k|\psi\rangle
([|\psi\rangle\langle \psi|,H])^{2k+1} |\psi\rangle = (- 1)^k V^k(E-H)|\psi\rangle
This starts looking like quantum signal processing, let's see more next.
Lemma 1. Any linear polynomial with a real root \(P(H) = 1-\tau H\) can be implemented this way.
([|\psi\rangle\langle \psi|,H])^{2k} |\psi\rangle = (- 1)^k V^k|\psi\rangle
([|\psi\rangle\langle \psi|,H])^{2k+1} |\psi\rangle = (- 1)^k V^k(E-H)|\psi\rangle
This starts looking like quantum signal processing, let's see more next.
Lemma 1. Any linear polynomial with a real root \(P(H) = 1-\tau H\) can be implemented this way.
Observation. If there is a complex root \(P(H) = H-z\) can be done by
\(U_\psi(P) = e^{i\theta |\psi\rangle\langle\psi|}e^{s[|\psi\rangle\langle\psi|,H]}\)
Result: Recursive iteration implements any QSP!
This starts looking like quantum signal processing, let's see more next.
Lemma 1. Any linear polynomial with a real root \(P(H) = 1-\tau H\) can be implemented this way.
4 stages of creating quantum algorithms
1. Design choice:
How to go about it?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
2. Unitary synthesis:
How to do it?
3. Circuit compilation:
What gates to do?
0. Problem choice:
What challenge to take up?
Double-bracket quantum algorithms:
Systematic framework for unitary synthesis
4 stages of creating quantum algorithms
Double-bracket algorithms with more details
By Marek Gluza
