Attested & literatureAlgorithmsQuantum linear algebra2026-08-12

Hybrid HHL++ for portfolio optimization

hhl algorithmlinear systemsportfolio optimizationhybrid quantum-classicaltrapped-ion hardware

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Adapt the Harrow-Hassidim-Lloyd (HHL) quantum linear-systems algorithm, most of whose components current noisy quantum hardware cannot reach, into a form that near-term devices can actually execute, and demonstrate it on an application. Yalovetzky, Minssen, Herman and Pistoia work on the gap between near-term-friendly proposals for the Harrow-Hassidim-Lloyd (HHL) algorithm — a quantum linear-algebra primitive whose components the abstract says are largely out of reach of noisy intermediate-scale quantum devices — and the circuits that can actually run on noisy hardware today. Building on the Hybrid HHL algorithm proposed by Lee and colleagues, the authors propose two modifications that together give their algorithm, Hybrid HHL++: a novel algorithm for choosing a scaling factor for the linear-system matrix that maximizes the use of the ancillary qubits allocated to HHL's phase-estimation component, and a heuristic for compressing the HHL circuit. They demonstrate the modified algorithm by running it on Quantinuum System Model H-series trapped-ion quantum computers, solving different instances of small-scale portfolio-optimization problems, which the abstract calls the largest experimental demonstrations of HHL for an application to date. The abstract frames this against the broader difficulty of application-oriented benchmarking of current quantum hardware, which it says the limited scale of most quantum-algorithmic demonstrations makes hard to perform.

Circuit & simulation

problem

encodetransform

algorithm

transformmeasure

readout

measure

What this takes and returns

TakesNothingWhat joins here

No input port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.

Nothing in the Atlas meets this end.

ReturnsNothingWhat joins here

No output port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.

Nothing in the Atlas meets this end.

This record publishes no gate sequence and no register, so there is nothing here to read an interface off. Absent rather than empty. See all 152 →

Where this sits

This record is named by the layer graph at:

HHL: eigenvalue inversion by phase estimation Method
Takes An access model for $A$ — sparse row/column entry oracles, or a block-encoding; a unitary preparing $\lvert b\rangle$ ; a known upper bound $\kappa$ on the condition number; the normalisation $\lVert A\rVert \le 1$ ; and a target state error $\varepsilon$ . Returns A flagged state $\varepsilon$ -close in $\ell_2$ to $A^{-1}\lvert b\rangle/\lVert A^{-1}\lvert b\rangle\rVert$ . It does not return $\lVert x\rVert$ , any entry of $x$ , or any classical functional of $x$ — those cost extra and are decided a layer above.

How it works

Yalovetzky, Minssen, Herman and Pistoia work on the gap between near-term-friendly proposals for the Harrow-Hassidim-Lloyd (HHL) algorithm — a quantum linear-algebra primitive whose components the abstract says are largely out of reach of noisy intermediate-scale quantum devices — and the circuits that can actually run on noisy hardware today. Building on the Hybrid HHL algorithm proposed by Lee and colleagues, the authors propose two modifications that together give their algorithm, Hybrid HHL++: a novel algorithm for choosing a scaling factor for the linear-system matrix that maximizes the use of the ancillary qubits allocated to HHL's phase-estimation component, and a heuristic for compressing the HHL circuit. They demonstrate the modified algorithm by running it on Quantinuum System Model H-series trapped-ion quantum computers, solving different instances of small-scale portfolio-optimization problems, which the abstract calls the largest experimental demonstrations of HHL for an application to date. The abstract frames this against the broader difficulty of application-oriented benchmarking of current quantum hardware, which it says the limited scale of most quantum-algorithmic demonstrations makes hard to perform. The Classiq library carries this subject under applications · finance. The sources read state no complexity bound for this record (The abstract of arXiv:2110.15958, the only source read for this record, states no complexity bound: no big-O expression, no run time, no qubit or gate count, and no scaling with the size of the linear system, the condition number, or any error parameter. What it states is a design contribution — "we propose two modifications to the Hybrid HHL algorithm proposed by Lee etal. leading to our algorithm Hybrid HHL++" — namely "a novel algorithm for determining a scaling factor for the linear system matrix that maximizes the utility of the amount of ancillary qubits allocated to the phase estimation component of HHL" and "a heuristic for compressing the HHL circuit" — and a hardware claim: "We demonstrate the efficacy of our work by running our modified Hybrid HHL on Quantinuum System Model H-series trapped-ion quantum computers to solve different problem instances of small-scale portfolio optimization problems, leading to the largest experimental demonstrations of HHL for an application to date." The Classiq index entry this record covers, applications/finance/hybrid_hhl_for_portfolio_optimization, gives a directory path and a file list and states no bound either. The field is therefore left empty on purpose rather than filled with a bound written from memory.).

Implementation

FrameworkUnsupported

hybrid-hhl-portfolio-optimization.txt

ALGORITHM: Hybrid HHL++ for portfolio optimization
PROBLEM: Adapt the Harrow-Hassidim-Lloyd (HHL) quantum linear-systems algorithm, most of whose components current noisy quantum hardware cannot reach, into a form that near-term devices can actually execute, and demonstrate it on an application.
IDEA: Yalovetzky, Minssen, Herman and Pistoia work on the gap between near-term-friendly proposals for the Harrow-Hassidim-Lloyd (HHL) algorithm — a quantum linear-algebra primitive whose components the abstract says are largely out of reach of noisy intermediate-scale quantum devices — and the circuits that can actually run on noisy hardware today. Building on the Hybrid HHL algorithm proposed by Lee and colleagues, the authors propose two modifications that together give their algorithm, Hybrid HHL++: a novel algorithm for choosing a scaling factor for the linear-system matrix that maximizes the use of the ancillary qubits allocated to HHL's phase-estimation component, and a heuristic for compressing the HHL circuit. They demonstrate the modified algorithm by running it on Quantinuum System Model H-series trapped-ion quantum computers, solving different instances of small-scale portfolio-optimization problems, which the abstract calls the largest experimental demonstrations of HHL for an application to date. The abstract frames this against the broader difficulty of application-oriented benchmarking of current quantum hardware, which it says the limited scale of most quantum-algorithmic demonstrations makes hard to perform.
REPORTED COST: Not stated by the sources read
BASIS: The abstract of arXiv:2110.15958, the only source read for this record, states no complexity bound: no big-O expression, no run time, no qubit or gate count, and no scaling with the size of the linear system, the condition number, or any error parameter. What it states is a design contribution — "we propose two modifications to the Hybrid HHL algorithm proposed by Lee etal. leading to our algorithm Hybrid HHL++" — namely "a novel algorithm for determining a scaling factor for the linear system matrix that maximizes the utility of the amount of ancillary qubits allocated to the phase estimation component of HHL" and "a heuristic for compressing the HHL circuit" — and a hardware claim: "We demonstrate the efficacy of our work by running our modified Hybrid HHL on Quantinuum System Model H-series trapped-ion quantum computers to solve different problem instances of small-scale portfolio optimization problems, leading to the largest experimental demonstrations of HHL for an application to date." The Classiq index entry this record covers, applications/finance/hybrid_hhl_for_portfolio_optimization, gives a directory path and a file list and states no bound either. The field is therefore left empty on purpose rather than filled with a bound written from memory.
DEMONSTRATED BY: the Classiq library entry applications/finance/hybrid_hhl_for_portfolio_optimization
PRIMARY SOURCE: Romina Yalovetzky, Pierre Minssen, Dylan Herman, Marco Pistoia (2021), Solving Linear Systems on Quantum Hardware with Hybrid HHL++ — https://arxiv.org/abs/2110.15958

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Quantum vs classical

Classical baseline

Compare Quantum linear algebra with the strongest classical method for the same instance, input budget, and output metric.

Quantum claim

This reference exposes a quantum circuit pattern; it does not imply an application-level speedup without a matched benchmark.

How to compare

Report input loading, circuit depth, repetitions, classical preprocessing, post-processing, and wall-clock time together.

Declared gaps

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Literature & references

Solving Linear Systems on Quantum Hardware with Hybrid HHL++2021 · Romina Yalovetzky, Pierre Minssen, Dylan Herman, Marco Pistoia

Primary source: it proposes the two modifications, the scaling-factor algorithm and the circuit-compression heuristic, that define Hybrid HHL++ on top of the Hybrid HHL algorithm of Lee and colleagues, and it reports running the modified algorithm on Quantinuum System Model H-series trapped-ion hardware to solve small-scale portfolio-optimization instances, calling this the largest experimental demonstration of HHL for an application to date. Consult it for how the scaling-factor algorithm and the compression heuristic actually work, for the size and structure of the portfolio-optimization instances solved, and for any complexity or resource analysis, none of which the abstract states.

arxiv.org/abs/2110.15958 ↗

Verification

Attested & literature

Peer-reviewed paper

Method: Curation only. The Classiq library publishes this subject as the directory `applications/finance/hybrid_hhl_for_portfolio_optimization`, pinned in scripts/classiq-parity/classiq-index.json; this record documents the algorithm that entry demonstrates, not the demonstration. The primary reference's title, authors and submission year were read from its arXiv abs page, and the complexity claim taken from a clause of that paper's abstract (The abstract of arXiv:2110.15958, the only source read for this record, states no complexity bound: no big-O expression, no run time, no qubit or gate count, and no scaling with the size of the linear system, the condition number, or any error parameter. What it states is a design contribution — "we propose two modifications to the Hybrid HHL algorithm proposed by Lee etal. leading to our algorithm Hybrid HHL++" — namely "a novel algorithm for determining a scaling factor for the linear system matrix that maximizes the utility of the amount of ancillary qubits allocated to the phase estimation component of HHL" and "a heuristic for compressing the HHL circuit" — and a hardware claim: "We demonstrate the efficacy of our work by running our modified Hybrid HHL on Quantinuum System Model H-series trapped-ion quantum computers to solve different problem instances of small-scale portfolio optimization problems, leading to the largest experimental demonstrations of HHL for an application to date." The Classiq index entry this record covers, applications/finance/hybrid_hhl_for_portfolio_optimization, gives a directory path and a file list and states no bound either. The field is therefore left empty on purpose rather than filled with a bound written from memory.). No circuit was constructed, simulated or run, and no Qmod model was executed.
Result: Pass · the record's problem statement and primary reference agree with the sources named above.
Caveat: This is a literature record. No circuit was built, compiled, simulated or run for it; no linear system was solved, no scaling factor was computed, and no circuit-compression heuristic was applied here. The record documents the algorithm the abstract describes and does not reproduce or verify the paper's own hardware demonstration. The Classiq index entry this record covers, applications/finance/hybrid_hhl_for_portfolio_optimization, gives only a directory path and a file list, nothing about their contents. The abstract states no complexity bound of any kind for Hybrid HHL++: no big-O expression, no run time, no qubit or gate count, and no dependence on the size of the linear system, the condition number, or any error parameter. What it states instead is a pair of design contributions — a method for choosing a scaling factor for the linear-system matrix, and a heuristic for compressing the HHL circuit — and a hardware result: the modified algorithm was run on Quantinuum System Model H-series trapped-ion quantum computers on different instances of small-scale portfolio-optimization problems. That hardware run is the paper's own reported result, at the scale it reports, and it was not repeated, re-run, or checked for this record; nothing here says what small-scale means in qubit count, circuit depth, or problem size, and nothing here evaluates the result against any later hardware or any later version of the paper. The claim that this is the largest experimental demonstration of HHL for an application to date is the paper's own comparison against the state of the art it knew of at the time, not a proven or independently checked ranking, and this record makes no claim about what has been demonstrated since. The abstract also builds on the Hybrid HHL algorithm proposed by Lee and colleagues without describing that prior algorithm, and this record does not describe or evaluate it either; nor does the abstract detail how the scaling-factor algorithm or the compression heuristic work, what properties they preserve, or what they cost — those specifics are in the paper, not here.

Facts

Record type: Literature reference
Classiq library entry: hybrid_hhl_for_portfolio_optimization
Reported cost: Not stated by the sources read
Classiq publication directory: applications/finance/hybrid_hhl_for_portfolio_optimization
Classiq shelf: applications · finance
Complexity basis: The abstract of arXiv:2110.15958, the only source read for this record, states no complexity bound: no big-O expression, no run time, no qubit or gate count, and no scaling with the size of the linear system, the condition number, or any error parameter. What it states is a design contribution — "we propose two modifications to the Hybrid HHL algorithm proposed by Lee etal. leading to our algorithm Hybrid HHL++" — namely "a novel algorithm for determining a scaling factor for the linear system matrix that maximizes the utility of the amount of ancillary qubits allocated to the phase estimation component of HHL" and "a heuristic for compressing the HHL circuit" — and a hardware claim: "We demonstrate the efficacy of our work by running our modified Hybrid HHL on Quantinuum System Model H-series trapped-ion quantum computers to solve different problem instances of small-scale portfolio optimization problems, leading to the largest experimental demonstrations of HHL for an application to date." The Classiq index entry this record covers, applications/finance/hybrid_hhl_for_portfolio_optimization, gives a directory path and a file list and states no bound either. The field is therefore left empty on purpose rather than filled with a bound written from memory.
Circuit: Not supplied

Source

Solving Linear Systems on Quantum Hardware with Hybrid HHL++ ↗

Kind: curated reference
Reviewed by: Leona Quantum curation pass
License: Citation metadata only; source publication terms apply