Double-bracket quantum signal processing

Marek Gluza

NTU Singapore

Scan QR code or go to slides.com/marekgluza

Z. Holmes

Y. Suzuki

B. Tiang

J. Son

N. Ng

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)

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 = (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

= - 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

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.

https://arxiv.org/abs/2504.01077

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.

https://arxiv.org/abs/2504.01077

Z. Holmes

Y. Suzuki

B. Tiang

J. Son

N. Ng

https://arxiv.org/abs/2504.01077

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

Design choices for optimization

Diagonalization		Inefficient?
Imaginary-time evolution		Non-unitary?
Quantum signal processing		Probabilistic?

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\|}

|\psi(z)\rangle = \frac{ p(H)|\psi\rangle}{\| p(H)|\psi\rangle\|}

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

Riemannian gradient: Unique vector \(g\) in the tangent space such that the directional derivative is a projection onto \(g\)

\nabla_A f(x) = \text{lim}_{s\rightarrow 0}\frac{ f(x+s A) - f(x)}{s} = \langle g, A\rangle

\nabla_A E(\psi) = \text{lim}_{s\rightarrow 0} \frac{ \langle \psi| e^{-isA}He^{isA}- H | \psi\rangle}{s} = \langle \psi| i [H,A] |\psi\rangle

Tr(ABC) = Tr(CAB)\\ \Downarrow \\ Tr(|\psi\rangle\langle \psi| [H,A] ) = Tr([|\psi\rangle\langle \psi|,H]A)

E(\psi) = \langle \psi| H | \psi\rangle

| \psi\rangle \mapsto e^{is A}|\psi\rangle

\nabla_A f(x) = \langle g, A\rangle

\nabla_A E(\psi) = \langle [H,|\psi\rangle\langle \psi|] , i 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 and uncovering new unitary synthesis formulas

Riemannian gradient: Unique vector \(g\) in the tangent space such that the directional derivative is a projection onto \(g\)

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),B],A(t)]

H \sim -i[A(t),B]

Heisenberg picture	Linear, variable: observable
Schroedinger picture	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 H(t)]]

Double-bracket quantum algorithms

are inspired by double-bracket flows

and allow for quantum compiling of short-depth circuits which approximate grounds states

How do you create a quantum algorithm?

Now: Not many distinct quantum algorithms
https://quantumalgorithmzoo.org/
Why?

00's Shor: Why have not more algorithms have been found? https://cs.brynmawr.edu/Courses/cs380/fall2012/Shor2003.pdf
Either because the easy ones are efficient classically or we have no good language

Classically: Reduction to decision problems
Quantumly: Is \(\langle\psi|H|\psi\rangle\) lower than ...?
Optimization: Only quadratic speed-up?

Grand unification of quantum algorithms https://arxiv.org/abs/2105.02859 through block-encodings
Another unification through double-bracket flows and Riemannian optimization - this talk
Double-bracket quantum algorithms give intuition why unification possible: \(\langle\psi|H|\psi\rangle\) is optimization so interpret through Riemannian gradients

Click links at slides.com/marekgluza

90's Shor: factoring is in BQP

\nabla_A f(x) = \langle g, A\rangle

\nabla_A E(\psi) = \langle [H,|\psi\rangle\langle \psi|] , i 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

What is the meaning of \(e^{s[|\psi\rangle\langle\psi|,H]}\)?

Riemannian gradient update
Quantum signal processing by a linear polynomial
Linear approximation to imaginary-time evolution

Fact 1. Riemannian gradient: Unique vector \(g\) in the tangent space such that the directional derivative is a projection onto \(g\)

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?

Double-bracket quantum algorithms:

Systematic framework for unitary synthesis

|\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)

|\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)\\ = e^{i\sqrt{\tau}H}e^{i\sqrt{\tau}|\psi\rangle\langle\psi|}e^{i\sqrt{\tau}H}|\psi\rangle + O(\tau^{3/2})

e^{itH}\\ e^{it|\psi\rangle\langle\psi|}

Product formula approximation:

\(e^{\tau[|\psi\rangle\langle\psi|,H]}| = e^{i\sqrt{\tau}H}e^{i\sqrt{\tau}|\psi\rangle\langle\psi|}e^{-i\sqrt{\tau}H}e^{-i\sqrt{\tau}|\psi\rangle\langle\psi|} + O(\tau^{3/2})\)

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?

|\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?

e^{itH}\\ e^{it|\psi\rangle\langle\psi|}

Two theorems 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:

1. Mean energy \(E_k := \langle \psi_k| H|{\psi_k}\rangle\) decreases as

\(E_{k+1} = E_k - 2s V_k + O(s^2)\)

where \(V_k := \langle \psi_k| H^2|{\psi_k}\rangle - E_k^2\)

2. Ground-state fidelity \(F_k := |\langle \lambda_0| {\psi_k}\rangle|^2\) increases as

\(F_{k} \ge 1 - q^k\)

where \(q = 1 - O( \Delta F_0 / \|H\|^3)\)

https://arxiv.org/abs/2412.04554

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:

https://arxiv.org/abs/2412.04554

Quantinuum

Double-bracket quantum algorithms

coherently implement Riemannian gradient steps
give rigorous unitary synthesis for
- imaginary-time evolution
- quantum signal processing
- diagonalization unitaries
Grover's algorithm is an approximation to imaginary-time evolution

J. Son

B. Tiang

S. Carrazza

R. Seidel

R. Zander

Z. Holmes

Y. Suzuki

Tell me when not fast enough? Get stuck? Something else?

Your input is needed to improve them!

N. Ng