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