Quantum Minimum Finding Techniques

• Quantum minimum finding techniques are a class of quantum algorithms that use amplitude amplification, phase estimation, and annealing to identify global or local minima in complex optimization landscapes.
• They extend Grover-type search methods with QRAM and iterative refinement, and adapt to noisy or approximate oracles while ensuring quadratic query speedup over classical methods.
• Practical implementations underpin advances in quantum machine learning, combinatorial optimization, and eigenvalue estimation, though challenges remain in hardware realization and error control.

Quantum minimum finding techniques constitute a central class of quantum algorithms designed to identify global or local minima in discrete or continuous optimization landscapes, leveraging quantum parallelism, amplitude amplification, quantum phase estimation, and quantum annealing paradigms. These methods yield (in several models) quadratic query complexity speedups over classical minimum finding, extend naturally to robust and oracle-imperfect regimes, and underpin applications ranging from combinatorial optimization to quantum machine learning.

1. Canonical Grover-Type Quantum Minimum Finding

Quantum minimum finding for a list of $N$ values achieved prominence with Dürr-Høyer-style algorithms, which adapt Grover's search to minimum selection. A representative instantiation uses Quantum Random Access Memory (QRAM) to coherently load all index-value pairs $|x\rangle\otimes|y_x\rangle$, enabling amplitude amplification over the relevant subspace (Albino et al., 2023):

• Oracle Construction: A phase-flip oracle $O_b$ marks $|x,y_x\rangle$ if $y_x<b$, realized by a multi-controlled $Z$ operation contingent on the binary comparison of $y_x$ with the classical bound $b$.
• Iterative Refinement: The algorithm iteratively learns the minimum by adjusting $b$ bitwise (most to least significant) over $m$ bits, updating $b \leftarrow y'$ whenever a strictly lower $y'$ is found.
• Time Complexity: $O(\sqrt{N})$ QRAM queries and $O(\mathrm{poly}(n+m))$ gate depth, with $n=\log N$, $m=\log(\max y)$. This is quadratically faster than the best classical $O(N)$ scan.
• Application: Used as an inner loop for quantum $k$-means clustering, reducing assignment overhead from $O(NK)$ to $O(N\sqrt{K})$ (Albino et al., 2023).

Efforts to guarantee success probability and gate-depth efficiency led to methods such as the optimized quantum minimum searching algorithm (OQMSA). OQMSA leverages the Grover-Long search operator for exact amplitude amplification, introducing phase-matched diffusion to achieve $P=1$ per subsearch and oracle circuit constructions of depth $O(n)$ (Liu et al., 2023). This strictly improves both query complexity and circuit resources over standard Grover-type approaches.

2. Quantum Approximate Minimum and $k$-Minimum Finding

Minimum finding in scenarios where only approximate or noisy value access is possible motivates robust and approximate quantum minimum algorithms (Gao et al., 2024, Quek et al., 2020). These extend the canonical model to accommodate $(\epsilon,\delta)$-approximate value oracles. Two central notions appear:

• Weak (k,$\epsilon$)-approximate minimum set: Each output $i_j$ satisfies $v_{s_j} \leq v_{i_j} \leq v_{s_j}+\epsilon$ for sorted true minima $v_{s_1}\leq\dots\leq v_{s_k}$.
• Strong (k,$\epsilon$)-approximate minimum set: $$\max_{i\in S} v_i \leq \min_{j\notin S} v_j+\epsilon$$.

The quantum algorithmic framework (Gao et al., 2024):

• Utilizes amplitude amplification/sampling and a generalized minimum-finding primitive.
• Maintains and hides already-found indices, combining with quantum counting to locate the $k$ smallest up to additive error $O(\epsilon)$.
• Query complexity is $\widetilde{O}(\sqrt{Nk})$ (strong and weak approximate sets), matching the exact-oracle case as $\epsilon,\delta\to0$.
• Applications encompass finding $k$ smallest expectation values of observables (using block encoding and amplitude estimation) and $k$ lowest eigenvalues of known-eigenbasis Hamiltonians with total queries $\widetilde{O}(\sqrt{Nk}/\epsilon)$ (Gao et al., 2024).

For noisy comparators (rather than value oracles), robust minimum finding compensates adversarial noise by recursively reducing the search space, leveraging Grover-based pivoting and classical $2$-approximation postprocessing, achieving $\tilde O(\sqrt{N(1+\Delta)})$ query complexity where $\Delta$ is the local density of ambiguous pairs (Quek et al., 2020).

3. Extensions: Quantum Annealing, Multistep, and Variational Techniques

Quantum annealing (QA) and related analog or hybrid techniques extend minimum finding to complex, high-dimensional or non-convex energy landscapes (Abel et al., 2021, Wang et al., 2023, Ismail et al., 2024):

• Quantum Annealing in Ising Landscapes: The system is initialized in an easily prepared ground state and adiabatically evolved under a time-dependent Hamiltonian from driver $H_\mathrm{initial}$ to problem Hamiltonian $H_\mathrm{problem}$ that encodes the cost function. QA exploits quantum tunneling to efficiently escape false minima that trap classical optimizers, showing empirically superior performance against thermal annealing, gradient descent, and Nelder-Mead in both discrete and continuous benchmarks (Abel et al., 2021).
• Multistep Quantum Resonant Transition: For discretized continuous minimization, a sequence of “threshold” Hamiltonians with nesting ground spaces $S_1 \supset S_2 \supset ...$ is constructed. At each step, the ground state is projected onto the subspace below a numerical threshold using quantum resonance. This exponentially contracts the search space, avoiding local traps and yielding deterministic location of global minima under mild spectral assumptions (Wang et al., 2023).
• QUBO and Quantum Annealing for Combinatorial Codes: Minimum distance problems for quantum stabilizer codes are recast as QUBO problems and mapped onto quantum annealers such as D-Wave Advantage. Scaling is $O(n\ln n)$ binary variables for code length $n$, and both pure QA and hybrid solvers are tested. While current QA hardware faces embedding and noise bottlenecks, the QUBO formulation provides a unifying interface for future improvements in quantum optimization hardware and algorithms (Ismail et al., 2024).

Variational Quantum Search (VQS) and Binary Search Hybrids: The Quantum Global Minimum Finder (QGMF) combines a classical binary search over an additive shift with a depth-$O(n)$ VQS circuit. By adaptively shifting the energy landscape, the algorithm isolates the minimum into a negative-value region, allowing low-depth quantum search for minima. For $n$ input bits, the query complexity is $O(n)$, outclassing Grover-type $O(\sqrt{N})$ scaling for near-term tasks (Soltaninia et al., 2024).

4. Minimum Eigenvalue and Energy Ground State Algorithms

The minimum-finding paradigm generalizes directly to Hermitian matrix or Hamiltonian eigenvalue problems—central in quantum chemistry and physics (Kerzner et al., 2023, Clausen et al., 2023, Chen et al., 2023):

• QPE + QAE Hybrid: For a Hermitian $H$, the minimum eigenvalue $\lambda_0$ is located via quantum phase estimation (QPE) and quantum amplitude estimation (QAE). Binary search, interleaved with QAE for thresholding, yields $\tilde{O}(\sqrt{N}/\epsilon)$ query complexity for dimension $N$ and error $\epsilon$. This provides a quadratic speedup over classical eigenvalue estimation and allows preparation of a state with substantial overlap in the lowest-energy subspace (Kerzner et al., 2023).
• Measurement-Based Feedback Control: An alternative measurement-based feedback scheme designs a control Hamiltonian (via semidefinite programming) and QND measurement loop to drive any initial mixed state to the minimum-energy eigenstate with almost sure convergence, providing a robust, parameter-free minimum-search method for diagonalizable operators (Clausen et al., 2023).
• Quantum Thermal Gradient Descent (QTGD): For local minima under physically motivated (thermal Lindbladian) perturbations, QTGD provably converges to an $\epsilon$-local minimum in $\mathrm{poly}(n,1/\epsilon)$ quantum time, a task classically hard under the same model for generic 2D Hamiltonians due to the equivalence to BQP-hard computation (Chen et al., 2023).

5. Resource Analysis, Oracle Models, and Practical Considerations

Quantum minimum finding techniques inherit significant dependence on oracle access, circuit depth, and NISQ compatibility:

Method	Query Scaling	Circuit Depth	Hardware Assumptions
Grover-type w/ QRAM	$O(\sqrt{N})$	polylog$(N)$	QRAM with $O(1)$ access, multi-ctrl gates
OQMSA	$O(\sqrt{N})$	$O(n)$	Optimized oracles, sure-success amplitude
QGMF (VQS+BS)	$O(\log N)$	$O(n)$	Low-depth VQS ansatz, NISQ-friendly
QA/QUBO	$O(n\ln n)$	Adiabatic, hardware	Annealer connectivity, penalty embedding
Approx. $k$-Min.	$\tilde O(\sqrt{Nk})$	polylog$(N,k,1/\epsilon)$	Composable $(\epsilon,\delta)$ oracles
Minimum Eigenvalue (QPE+QAE)	$\tilde O(\sqrt{N}/\epsilon)$	$O(\sqrt{N}/\epsilon)$	Block-encoding, controlled evolution

Practical scalability is often bottlenecked by QRAM construction, multi-controlled-gate decompositions, and hardware decoherence, particularly for fault-tolerance. Variational and annealing methods offer lower-depth alternatives, but their success may depend on cost-function structure, error-mitigation, and the presence or absence of barren plateaus or embedding constraints.

6. Applications and Extensions

Quantum minimum finding subroutines are foundational in:

• Machine learning primitives (e.g., quantum $k$-means, $k$-nearest neighbor classification) (Albino et al., 2023, Miyamoto et al., 2019)
• Ground-state and thermal-state preparation in quantum many-body systems (Kerzner et al., 2023, Chen et al., 2023)
• Quantum error correction and code certification via minimum distance calculation (Ismail et al., 2024)
• Continuous and combinatorial optimization in non-convex energy landscapes (Abel et al., 2021, Wang et al., 2023, Soltaninia et al., 2024)
• Robust hypothesis selection with noisy/comparator-limited quantum access (Quek et al., 2020)

Extensions under consideration include robust/search variants using untrusted or bounded-error oracles, minimum-finding for weighted or streaming data, parallel minimum search, and integration into composable subroutines for higher-level quantum algorithms (e.g., quantum SDP solvers, multi-Gibbs sampling, matrix games).

7. Current Limitations and Future Directions

Despite the quadratic quantum speedup in query complexity, several bottlenecks remain:

• Physical QRAM with true $O(1)$ access is an open engineering challenge (Albino et al., 2023).
• Multi-controlled operations and long coherence times remain expensive; error-corrected quantum devices are necessary for large database sizes.
• Classical overheads for data loading, oracle construction, and post-processing can offset asymptotic scaling.
• For hybrid QA/QUBO methods, embedding overhead and noise currently make classical solvers preferable at moderate size, but improved hardware may alter this calculus as connectivity and coherence increase (Ismail et al., 2024).
• Minimum finding with approximation/robustness constraints is sensitive to the oracle/measurement model, with several competing approaches demonstrating distinct tradeoffs in complexity vs. error guarantees (Gao et al., 2024, Quek et al., 2020).

Exploration of near-term viability (e.g., QGMF for NISQ devices (Soltaninia et al., 2024)) and further reduction in required circuit depth or query complexity remain active areas. Algorithmic innovations in error-mitigation, quantum memory construction, and compositional algorithm design are key to expanding the applicability of quantum minimum finding techniques across scientific and industrial problems.