Quantum Application Solvers

Integrate quantum computing
into your application workflows

Work with problem-specific classical inputs and outputs, while hybrid quantum-classical solvers handle the computation in between.

Explore the problemDiscover Hilvarion™

01 / The application problem

Define the answer
before choosing the algorithm

Application solvers sit above circuit programming. For combinatorial optimisation, the input is a mathematical problem: a set of decisions, a cost to minimise and conditions that determine which answers are allowed.

This is different from quantum compilation. A compiler implements a given computation on a target device. An application solver chooses and runs a computational procedure to search for an answer to the original problem.

01

Decision variables

Specify what an assignment means in the application, rather than treating a measured bitstring as an answer without interpretation.

02

Objective and feasibility

State which costs matter and which assignments are valid. Encoding constraints can change the quantum resources required.

03

Evaluation budget

Define the available circuit evaluations, measurements and classical work before comparing solution methods.

A binary optimisation problem

x⋆∈argmin⁡x∈F  C(x),F⊆{0,1}nx^\star\in\underset{x\in\mathcal F}{\operatorname{argmin}}\;C(x),\qquad\mathcal F\subseteq\{0,1\}^n

Here x is a candidate assignment, C is its cost and the feasible set contains the allowed assignments. The objective defines how solutions are ranked, while the feasible set expresses the conditions an assignment must satisfy.

Problem formulation / Weighted Max-Cut

C(x)=−∑(i,j)∈Ewij(xi⊕xj)C(x)=-\sum_{(i,j)\in E}w_{ij}(x_i\oplus x_j)
HC=−12∑(i,j)∈Ewij(I−ZiZj),HC∣x⟩=C(x)∣x⟩H_C=-\frac12\sum_{(i,j)\in E}w_{ij}(I-Z_iZ_j),\qquad H_C|x\rangle=C(x)|x\rangle

Each bit assigns a vertex to one of two groups. The XOR operation is one when an edge’s endpoints differ. Minimising the negative cut is equivalent to maximising the weight between the groups. The corresponding diagonal Hamiltonian gives each basis state the same cost.

02 / The hybrid computation

Quantum sampling
inside a classical search

In QAOA, alternating cost and mixing operations define a parameterised distribution over candidate bitstrings. A classical procedure evaluates the measured outcomes and proposes new circuit parameters. The angles are controls for the search, not the application’s answer.

Hilvarion combines an adaptive initial state with this variational loop. Measured solutions inform a bias between optimisation stages, while the cost and mixer angles are optimised within each stage.

Read the QAOA formulation
  1. 01problem

    Formulate

    Specify variables, an objective and any feasibility conditions. Preserve the original instance.

  2. 02model

    Encode

    Construct a problem Hamiltonian and the circuit family used to explore candidate solutions.

  3. 03circuit

    Prepare

    Choose parameters and, for hardware execution, compile the circuit for a defined target.

  4. 04execution

    Sample

    Repeat circuit execution to collect bitstrings. Evaluate the original objective on observed outcomes.

  5. 05feedback

    Update

    Use a classical score to update parameters and preserve the best candidate solutions found.

  6. 06evidence

    Evaluate

    Assess solution quality and resources. Report classical refinement separately from raw sampling.

Sampling and classical updates repeat within a resource budget. Weylattice compiles the circuits for the selected target, while Spinorio Workflow records the associated execution.

Multi-layer adaptive QAOA / Algorithmic example

∣ψp(θ,γ,β)⟩=[∏ℓ=1p←e−iβℓHMe−iγℓHC][⨂j=1nRy(θj)]∣0⟩⊗n|\psi_p(\boldsymbol{\theta},\boldsymbol{\gamma},\boldsymbol{\beta})\rangle=\left[\overleftarrow{\prod_{\ell=1}^{p}}e^{-i\beta_\ell H_M}e^{-i\gamma_\ell H_C}\right]\left[\bigotimes_{j=1}^{n}R_y(\theta_j)\right]|0\rangle^{\otimes n}
θj=π2−(−1)(bT)jδ\theta_j=\frac{\pi}{2}-(-1)^{(b_T)_j}\delta

A biased initial state is followed by p alternating cost and mixer layers, each with its own parameters. Classical optimisation adjusts these parameters, while measurement feedback updates the initial-state bias between optimisation stages.

The left-pointing product orders the layers so that layer 1 acts first, on the right. State preparation is applied once, before the p layers. Each layer has its own gamma and beta, giving 2p cost-and-mixer angles. The mixer HM is the sum of single-qubit X operators.

The preparation angles theta are set by the target bitstring bT and bias delta, then held fixed within each optimisation stage. Zero bias recovers the uniform initial state. Circuit depth p and the number of classical optimisation stages are separate choices.

Two kinds of feedback

A circuit score guides the angle optimiser. Separately, the best measured bitstring updates the search record and can inform the next stage’s preparation. A good circuit score and a newly discovered solution are not the same event.

CVaR scores a selected low-cost fraction of the outcomes. Its tail fraction and tie-handling policy must be recorded because different output distributions can receive the same score.

CVaR and variational optimisation
Quantum work
Prepare circuits and sample their measurement outcomes.
Classical work
Score candidates, update parameters, retain solutions and optionally refine them.
Execution work
Compile and schedule circuits for a target, then assess hardware errors separately from algorithm design.

03 / The evidence

Finding a solution
is not the same as repeating it

A biased state can repeatedly produce an already-known solution. That does not establish that quantum dynamics helped discover it. Solver evaluation must distinguish search progress from output concentration and classical post-processing.

Comparisons should use explicit resource budgets, multiple instances and independent evaluation samples. A small simulated example cannot establish a useful performance advantage over classical optimisation.

A controlled question / Can the circuit improve a fixed target?

Pimprove=Pr⁡X∼Pθ,γ,β[C(X)<C(bT)]P_{\mathrm{improve}}=\Pr_{X\sim P_{\boldsymbol{\theta},\boldsymbol{\gamma},\boldsymbol{\beta}}}[C(X)<C(b_T)]

Compare preparation-only, mixer-only and cost-plus-mixer circuits from the same target and bias. This diagnostic isolates one part of the search; the complete solver still needs end-to-end comparisons with classical methods.

01

Quality of discovered solutions

Record the best objective value found and when it was reached. Report optimality gaps only when a valid reference optimum or bound is available.

02

Resources needed to reach it

Count circuit evaluations and shots alongside classical computation and elapsed time. Matching shot counts alone does not equalise total computational cost.

03

Contribution of each layer

Keep algorithm design, compilation, error suppression and classical refinement independently configurable so their effects can be tested.

04

Independent evaluation

Freeze the selected circuit before collecting fresh evaluation samples. Preserve raw outcomes separately from any classically improved solutions.

Evaluate the complete computation

Compare solution quality and time to a target against relevant classical methods. Include formulation, circuit execution and classical refinement in the resource budget, rather than evaluating the quantum circuit in isolation.

04 / Solve with Hilvarion™

Problem-specific inputs,
evaluated solutions

Hilvarion combines problem formulation, hybrid quantum-classical optimisation and solution evaluation in a single workflow. Submit your problem data, configure the solver and receive candidate solutions with their objective values. Inspect the associated circuits and execution records through Spinorio Workflow.

Hilvarion brings application-level problem solving into Spinorio Lab. Weylattice handles quantum circuit compilation, while Spinorio Workflow records the circuits, jobs and results associated with each solver run.

An integrated application workflow

01 / Problem configuration

Formulation and search

Supply problem data, configure the algorithm and set the evaluation budget. Hilvarion connects the problem model to the hybrid search.

02 / Platform integration

Compilation and execution

Run the solver in Spinorio Lab, with Weylattice compiling its circuits and Spinorio Workflow recording the associated jobs and results.

03 / Solution evaluation

Results and provenance

Inspect candidate solutions, objective values and resource use. Keep raw measurements distinct from classical refinement and retain the execution context.

HILVARION™ / Quantum Application Solvers

Integrate Hilvarion into your workflow

Explore the solver capabilities or discuss your application, its problem formulation and the criteria used to evaluate solutions.

Explore HilvarionDiscuss an application

Selected references