QILaplace.jl

Quantum-inspired signal transforms via tensor-network representations.

What this package provides • Core features • Quickstart • Performance • Authorship

QILaplace.jl is a Julia package for Fourier- and Laplace-style transforms using tensor networks on classical hardware. It is built for large, structured signals, where compressed MPS/MPO representations keep these transforms practical when dense workflows become too expensive.

The package includes QFT, damping-transform (DT), and full z-transform (zT) pipelines, together with MPS compression, MPO construction, and direct coefficient extraction tools.

What this package provides

• Signal-to-MPS encoding using SVD and randomized SVD (:svd / :rsvd).
• Compressed transform MPOs for QFT, DT, and zT.
• Scalable transform workflows for compressible signals, with practical near-logarithmic growth in problem size.

DT and zT use the same high-level API style as QFT, so moving between Fourier and Laplace/z-domain analyses is straightforward.

Core features

1) Signal compression workflows (SVD and RSVD)

• Convert dense vectors into MPS (signal_mps, signal_ztmps)
• Control approximation with cutoff and maxdim
• Choose deterministic SVD (sequential sweep) or randomized SVD (divide and conquer) backends

2) Circuit compression for transform MPOs

• Build compressed transform operators with build_qft_mpo, build_dt_mpo, and build_zt_mpo
• Apply through the same contraction interface (W * ψ)
• Keep transform accuracy under explicit truncation control
• The circuit-compression construction is strictly optimal for these transform circuits.

For full circuit diagrams and examples, see:

• Discrete Fourier Transform
• Damping Transform
• Complex Laplace Transform

3) z-domain analysis and pole scanning

• Query transformed coefficients directly from tensor states (coefficient)
• Run coarse-to-fine scans in $(k,\ell)$, $s$-, and $z$-spaces
• Compare inferred peak/pole locations against analytical references

See the full pole-identification walkthrough in:

• https://SUTD-MDQS.github.io/QILaplace.jl/tutorials/zt/

Workflow

1. Encode signal -> signal_mps (single register) or signal_ztmps (paired register).
2. Build transform MPO -> build_qft_mpo, build_dt_mpo, or build_zt_mpo.
3. Apply + extract coefficients -> W * ψ (or apply(W, ψ)) and probe entries with coefficient.

Quickstart tour

Signal -> MPS -> QFT frequency bins

using QILaplace

n = 10                                            # log2 signal length -> 1024 samples
signal = generate_signal(n; kind=:sin_decay,
                         freq=[1.0, 2.5], decay_rate=[0.08, 0.03])

ψ = signal_mps(signal; method=:rsvd, cutoff=1e-9, maxdim=64)
compress!(ψ; maxdim=64)

Wqft = build_qft_mpo(ψ; cutoff=1e-12, maxdim=128)
spectrum = Wqft * ψ                               # Apply MPO to MPS

# Extract physical-scale amplitude at frequency bin (bit-reversed order)
amplitude = coefficient(spectrum, "0101010110")

Damped and z-transform workflows (paired register)

using QILaplace

n = 10
signal = generate_signal(n; kind=:sin_decay, freq=1.0, decay_rate=0.05)

ψzt = signal_ztmps(signal; method=:svd, cutoff=1e-12)

# Damping transform (real-axis)
Wdt = build_dt_mpo(ψzt, 0.3; maxdim=64)
damped = Wdt * ψzt

# Full z-transform (DT + QFT circuit in one MPO)
Wzt = build_zt_mpo(ψzt, 0.3; maxdim=128)
response = Wzt * ψzt

amplitude_dt = coefficient(damped, "1010101010")
amplitude_zt = coefficient(response, "1010101010")

Performance

For a signal of length $N = 2^n$, transforms are applied as MPO-MPS contractions over $n=\log(N)$ tensor sites. With signal bond dimension $\chi_s$, circuit bond dimension $\chi_c$, and bounded circuit bond behavior for fixed accuracy:

$$ T_{\mathrm{apply}} \sim O\left( \chi_s^2 \log N \right). $$

For full benchmarks and runtime plots, see: https://SUTD-MDQS.github.io/QILaplace.jl/dev/benchmarking/

Authorship and maintenance

• Lead maintainer: Gauthameshwar S.
• Original codes: Noufal Jaseem.
• PI: Dario Poletti.

Made with ❤️ using Julia, ITensors, and a healthy respect for bond-dimension growth.

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