qsp-control-quantum

name	qsp-control-quantum
description	Quantum Signal Processing (QSP) framework for analytical quantum control of qubit-oscillator systems. Use when designing quantum control protocols, mitigating cross-Kerr interactions, constructing Fock-state-selective operators, or mapping control problems to QSP form. Triggers: QSP control, quantum signal processing control, qubit-oscillator control, Fock state manipulation, cross-Kerr mitigation, analytical quantum control, Jaynes-Cummings QSP.
metadata	{"arxiv_id":"2606.26085","published":"2026-06-24","authors":"Aishwarya Majumdar, John M. Martyn, Yuan Liu, Nathan Wiebe","tags":["quantum","control-theory","signal-processing","qubit-oscillator","Fock-states"]}

QSP-Control: Analytic Quantum Control via Quantum Signal Processing

Core Concept

Map quantum control problems to the Quantum Signal Processing (QSP) formalism, enabling analytical (not heuristic) design of control pulses with rigorous error guarantees. QSP provides a systematic framework for implementing unitary and non-unitary transformations, replacing brute-force pulse optimization with constructive polynomial methods.

Applicability

• Dispersively coupled qubit-oscillator systems
• Mitigating unwanted nonlinear effects (cross-Kerr interactions)
• Precise Fock state manipulation
• Any quantum control problem where target operations can be expressed as polynomial functions

QSP-Control Framework

Step 1: Identify the Control Problem

Express the quantum system Hamiltonian in the standard form:

• System: qubit-oscillator with dispersive coupling
• Unwanted terms: cross-Kerr nonlinearities, leakage
• Target: specific unitary evolution or state preparation

Step 2: Map to QSP Formalism

The key structural insight: Jaynes-Cummings interaction has the same mathematical structure as QSP rotations.

Jaynes-Cummings:  H_JC = g(a†σ⁻ + aσ⁺)
QSP rotation:     R(θ) = exp(-i θ/2 σ_x)

This structural parallel enables:

1. Encoding oscillator number states into QSP rotation angles
2. Constructing arbitrary polynomial functions of photon number
3. Achieving number-selective operations without numerical optimization

Step 3: Construct Fock-State-Selective Operators

For targeting specific Fock states |n⟩:

1. Identify target polynomial: Define P(n) that equals 1 for target states, 0 otherwise
2. Use QSP phase angles: Compute QSP phases {φ_k} that implement P(n) via alternating rotations
3. Apply Jaynes-Cummings mapping: Each QSP rotation → evolution under JC Hamiltonian for specific time

The resulting sequence implements the desired Fock-state-selective operation analytically.

Step 4: Mitigate Cross-Kerr Effects

Cross-Kerr interactions (χ a†a σ_z) cause unwanted phase accumulation. QSP-Control approach:

1. Express the cross-Kerr evolution as a QSP signal operator
2. Design QSP sequence that applies identity on the cross-Kerr term while implementing target operation
3. The QSP framework guarantees cancellation to arbitrary precision

Step 5: Error Analysis

QSP provides built-in error guarantees:

• Approximation error bounded by polynomial degree
• Robustness to parameter errors characterized analytically
• Gate fidelity scales predictably with sequence length

Key Mathematical Tools

• Chebyshev polynomial decomposition: Express target functions as Chebyshev series
• QSP phase computation: Algorithms to find phases {φ_k} for target polynomial
• SU(2) signal operator representation: Map system dynamics to alternating rotations

Implementation Workflow

Problem → Hamiltonian analysis → QSP mapping → Polynomial specification
  → Phase computation → Pulse sequence generation → Error verification

1. Analyze system Hamiltonian, identify controllable terms
2. Map dynamics to SU(2) signal operator form
3. Specify target polynomial (what transformation to achieve)
4. Compute QSP phases (use existing QSP phase-finding algorithms)
5. Translate phases to physical pulse parameters (durations, amplitudes)
6. Verify error bounds analytically

Pitfalls

• QSP requires SU(2) structure: Not all control problems map naturally; may need effective two-level approximation
• Phase computation complexity: Finding QSP phases for high-degree polynomials can be numerically challenging
• Decoherence limits: Long QSP sequences increase exposure to decoherence — balance polynomial degree with coherence time
• Cross-Kerr strength: Very strong cross-Kerr may break the perturbative assumptions used in QSP mapping

Related Skills

• quantum-signal-processing-orthogonal-polynomials — QSP mathematical foundations
• quantum-control-engineering — General quantum control patterns
• quantum-control-pulse-software — Pulse-level quantum control software
• jaynes-cummings-oscillator-control — JC oscillator control methods