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