Multi-Objective Decision-Making: Classical to Quantum Paradigm Mapping#

Application Scenarios#

Classical Multi-Objective Decision-Making#

Any problem that requires simultaneous optimization across multiple objective dimensions, for example:

  • Product design: Balance performance, cost, reliability, manufacturability

  • Investment portfolio: Weigh returns, risk, liquidity, compliance

  • Policy making: Consider economic effects, social fairness, environmental costs

Quantum Multi-Objective Decision-Making#

Problems that require quantum computing to simultaneously optimize multiple objectives encoded as Hamiltonian operators, for example:

  • Encode multiple optimization objectives as different Hamiltonian operators

  • Quantum encoding and simultaneous satisfaction of multiple physical constraint conditions

  • Quantifiable observable of multiple performance indicators in complex systems

Core Paradigm#

Core Elements#

Key Characteristic: What is evaluated is not the characteristics of the strategy itself, but the achievement of the strategy on the problem’s objective dimensions.

Classical Element

Quantum Mapping

Role Description

Objective Dimensions

Hamiltonian Observable

Defines multiple aspects of the problem to optimize

Strategy Scheme

Trial Circuit Architecture

Different solving methods

Evaluation Result

Observable Expectation Value

Performance of the strategy on each objective

Core Process#

1. Define Objective Dimensions#

  • Classical: Identify multiple inherent objective dimensions of the problem

  • Quantum: Determine target observables

    • Select a set of Hamiltonian operators {Ĥ₁, Ĥ₂, …, Ĥₘ} representing key dimensions of the problem

2. Generate Strategy Schemes#

  • Classical: Design different implementation strategies

  • Quantum: Prepare trial circuit strategies

    • Design multiple quantum circuit architectures {U₁, U₂, …, Uₙ}

3. Multi-Dimensional Evaluation#

  • Classical: Evaluate each strategy’s performance on all objective dimensions

  • Quantum Multi-Dimensional Evaluation:

    • For each circuit Uᵢ, optimize parameters to obtain quantum state |ψᵢ⟩

    • For each state |ψᵢ⟩, measure expectation values of all m observables

    • Obtain evaluation vector: Vᵢ = [⟨Ĥ₁⟩ᵢ, ⟨Ĥ₂⟩ᵢ, …, ⟨Ĥₘ⟩ᵢ]

4. Strategy Comparison and Selection#

  • Classical: Analyze the strength and weakness distribution of each strategy, select based on situational needs

  • Quantum:

    • Compare performance of different circuits on each objective

    • Identify advantage dimensions of each strategy

    • Select appropriate circuit based on problem priorities

Value Demonstration#

  • Systematic Strategy Evaluation: Examine strategy performance on multiple related objectives simultaneously

  • Reveal Strategy Preferences: Understand which objective dimensions different circuit architectures tend to optimize

  • Contextualized Selection: Choose the most suitable quantum algorithm based on specific problem requirements

  • Avoid Single-Dimension Bias: Prevent selecting strategies based solely on single indicators (such as energy)