Optimize Your Portfolio with the Harrow–Hassidim–Lloyd algorithmTransparent, Reproducible Results
Benchmark portfolio allocation with reproducible HHL outputs.
Build Efficient Portfolios from Your Data
Mean-variance portfolio optimization
You provide historical asset prices. From these we compute expected returns r and the covariance matrix \Sigma. The goal is to find allocation weights w that achieve a target return (\mu) with minimal risk under standard constraints (weights sum to 1). Formally:
To solve it efficiently, we rewrite the optimization as a linear system Ax = b, which collects variables and constraints into one compact problem.
- • End-to-end HHL portfolio optimization (objective and constraints) executed on a simulator.
- • Results: allocation weights w, portfolio risk w^\top \Sigma w, feasibility & constraint checks — seed-controlled and reproducible.
- • Downloadable, research-paper–style report (method, experiments, results, references) — citable.
- • Executable code in Python you can run as-is, with seeds and logs.
- • Quantum-hardware implementation details: data encoding, circuit design, qubit/depth estimates, and backend/cost guidance.
- • Benchmark-aligned classical baseline comparison with metrics and plots.
Harrow–Hassidim–Lloyd: The Quantum Linear-System Solver
HHL (Harrow–Hassidim–Lloyd) solves Ax = b and—under standard assumptions—offers potential exponential speedups when the goal is to estimate functionals of the solution (e.g., portfolio-level risk \langle x|M|x\rangle) rather than reconstruct every component of x. Today on our platform HHL runs in simulation, so you receive the full allocation weights and constraint verification with high accuracy in a reproducible workflow.
Strengths (Hypotheses)
- Potential exponential speedup (in N): When A is sparse, Hermitian, well-conditioned—and |b\rangle preparation and e^{iAt} are efficient—HHL can offer exponential-scale advantages for estimating functionals of the solution (e.g., \langle x|M|x\rangle).
- Bypasses readout bottlenecks: HHL prepares |x\rangle and lets you measure decision metrics directly, avoiding full vector reconstruction.
- Efficient for many "what-if" runs with fixed A: If only the right-hand side b changes, you can re-evaluate functionals of the solution while reusing the same A handling—ideal for scenario analysis.
- Improves with classical preconditioning: Conditioning/regularizing A to better meet HHL assumptions can boost accuracy and success probability without altering the quantum core.
Weaknesses & Risks
- Scope of speedup: The theoretical speedup applies to limited information (functionals like \langle x|M|x\rangle), not to recovering the full vector.
- Input & simulation requirements: To retain polylogarithmic scaling, |b\rangle must be efficiently encoded and e^{iAt} efficiently simulated—practical mainly when A is sparse and well-conditioned.
- Parameter sensitivity: in simulation we may use classically computed min/max eigenvalues to tune parameters for high accuracy; at scale we rely on estimated bounds, which can reduce accuracy.
- Hardware maturity: Current quantum hardware is not yet ready for reliable end-to-end HHL at scale; our delivery today is high-accuracy simulation, with hardware benefits viewed as forward-looking.
Proof-of-Concept Simulation Results
Task: Mean-variance portfolio optimization on 3 assets; target return 2.1%/mo; weights sum to 1.
Execution
Simulator
Qubits
13
Training
Not needed
Key Outcomes
Expected Return
\approx 2.06\%/ month
Risk
\approx 0.43\%/ month
Feasibility (return constraint)
\|r^\top w - \mu\| \approx 2.0 \times 10^{-6}
Budget
\|\mathbf{1}^\top w - 1\| \approx 1.0 \times 10^{-3}
Classical Residual
\|Ax - b\| \approx 6.2 \times 10^{-5}
Business Impact
As fault-tolerant quantum hardware becomes available in the coming years, HHL-based portfolio optimization will unlock transformative business advantages:
Computational Efficiency
Fewer computational steps per scenario at scale (N ~ 10⁶–10⁸)
When efficient state preparation and time evolution are implemented
Scenario Capacity
More optimization runs for different scenarios per day
Accelerating portfolio strategy testing and validation
Cost Optimization
Lower computational spending per day
Getting significantly more value from the compute budget.
How it works
Simple and transparent: from your brief to quantum results, code, and a paper
Describe
Describe your problem in plain language; map it to the right use case
Confirm
Confirm a domain-adapted quantum/hybrid method and key assumptions
Run
Download the ready-to-run code; execute on a simulator with a fixed seed
Review
Review reproducible results and logs — iterate in the Run → Review loop
Benchmark
Compare against a classical baseline; run on quantum hardware
Run HHL Now
Run your first HHL portfolio optimization and get transparent, reproducible results in minutes.
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