Qubit-Efficient Encoding Methods

• Qubit-efficient encoding is a method that exploits mathematical and structural properties to map classical data or quantum states onto fewer qubits, significantly reducing resource requirements in NISQ devices.
• Techniques such as Matrix Product State mappings, edge-selection for QAOA, and Gray code encodings minimize qubit use by optimizing data permutations and leveraging problem-specific constraints.
• These approaches enable scalable applications in quantum machine learning, Hamiltonian simulation, and combinatorial optimization while balancing trade-offs between circuit depth, error control, and computational overhead.

Qubit-Efficient Encoding

Qubit-efficient encoding comprises techniques that minimize the number of qubits required to faithfully represent classical data, objective functions, or quantum states on a quantum computing device. Instead of naively mapping every degree of freedom to an individual qubit, these approaches exploit mathematical, statistical, or structural properties of data and quantum algorithms to maximize information density per qubit. The central goals are (i) reducing resource requirements for NISQ and near-term quantum devices, (ii) maintaining high fidelity and efficient circuit depth, and (iii) enabling scalable quantum simulation and optimization for real-world datasets and Hamiltonians.

1. Matrix Product State (MPS) and Optimal Qubit Mapping

Matrix Product State (MPS) representations provide a tensor-network-based formalism for encoding high-dimensional classical data into the amplitudes of quantum states, allowing approximate representations with a number of parameters scaling as $O(N\chi^2)$, where $N$ is the number of qubits and $\chi$ the bond dimension. Given a normalized real vector $x\in\mathbb{R}^{2^N}$, amplitude encoding associates $$x \mapsto |\psi\rangle = \sum_{i=0}^{2^N-1} a_i|i\rangle$$. In MPS-based encoding, the tensor $T^{s_1\dots s_N}$ of amplitudes is approximated by a product of local 3-way tensors, and the quantum state becomes

$$|\psi_{\rm MPS}\rangle =\sum_{s_1\cdots s_N} \Bigl(\prod_{k=1}^N A^{[k]\,s_k}\Bigr)\, |s_1\cdots s_N\rangle,$$

with each $A^{[k]}\in\mathbb{R}^{\chi\times2\times\chi}$.

To optimally exploit qubits, one can permute the mapping of classical data bits to qubit indices to minimize the truncation error from SVD at each MPS bipartition:

$$\varepsilon^2(\pi)= \sum_{k=1}^{N-1}\sum_{j=\chi+1}^{r_k}(\lambda_j^{(k)})^2 = \|\psi_\pi-\psi_{\pi,\mathrm{MPS}}\|^2.$$

A uniform-cost search algorithm with symmetry pruning identifies data-dependent optimal permutations, reducing Frobenius errors by 23–32% across a range of bond dimensions and yielding classifier fidelity gains up to $+0.4\%$ at $N$0 (Jeon et al., 2024).

Recommended practice is to select the minimal $N$1 consistent with fidelity targets and to always perform a permutation search before SVD—even for $N$2, the overhead is negligible: tens of seconds. This approach, combined with hardware-aware circuit compilation and flexible SWAP networks, enables scalable, qubit-efficient amplitude encoding for quantum machine learning and data-driven algorithms.

2. Qubit-Efficient Formulations for Combinatorial Optimization

Classical quadratic unconstrained binary optimization (QUBO) and combinatorial problems such as TSP or MaxCut typically require $N$3 or worse in qubits with standard one-hot or unary encodings. Several recent advances provide systematic qubit reductions:

• Edge-Selection Encoding for QAOA: By encoding traveling salesperson tours with edge selection variables $N$4 (1 if edge is present in the tour, 0 otherwise) instead of 1-hot variables $N$5, the number of qubits is reduced from $N$6 to $N$7. Constraint handling is shifted into quantum mixer layers or handled via quadratic penalties, and the number of two-qubit gates is suppressed to $N$8 per QAOA layer (compared to $N$9 for the 1-hot encoding). This method exhibits improved empirical approximation ratios (mean relative error reduced from $\chi$0 to $\chi$1 at $\chi$2) at the cost of a modest increase in optimizer iterations (Garhofer et al., 2024).
• Exponential Penalty Encoding: Quadratic-constraint QUBOs with slack variables and additional ancilla qubits can be replaced by exponential penalties $\chi$3 for each constraint $\chi$4. Expanding up to quadratic order leverages $\chi$5 for binary variables, yielding a penalty that is strictly quadratic in $\chi$6 and does not introduce any extra variables. Applications to the Bin Packing Problem and TSP achieve qubit count reductions of $\chi$7 and $\chi$8, respectively, with empirical solution quality (e.g., $\chi$9 approximation probability for TSP with $x\in\mathbb{R}^{2^N}$0 qubits) comparable to or exceeding slack-based approaches (Kantbekova et al., 9 Sep 2025).
• Factoring and Gray Code Mappings: For certain problems, the solution space is intrinsically smaller than the Boolean hypercube. Factoring-based encodings, Gray-code traversals, and index/value mappings permit the encoding of $x\in\mathbb{R}^{2^N}$1 classical variables into $x\in\mathbb{R}^{2^N}$2 qubits with increasing expressivity up to $x\in\mathbb{R}^{2^N}$3-body variable correlations (Tan et al., 2020, Matteo et al., 2020).

3. Qubit-Efficient Encodings in Electronic Structure and Hamiltonian Simulation

Hamiltonian simulation of fermionic systems—central to quantum chemistry—traditionally employs Jordan–Wigner or Bravyi–Kitaev mappings with one qubit per spin-orbital, often populating a small fraction of the resulting $x\in\mathbb{R}^{2^N}$4 Hilbert space. Several strategies have been developed to address this inefficiency:

• Configuration-Space Encodings (QEE): By mapping only physical configurations corresponding to fixed particle number $x\in\mathbb{R}^{2^N}$5 (and possibly additional symmetries, such as total spin) and using lexicographical combinatorial indexings, the required qubit count drops to $x\in\mathbb{R}^{2^N}$6. The encoded Hamiltonian is constructed as a sum of Pauli strings via entry-operator expansion, and variational quantum eigensolvers (VQE) based on this mapping show quantitative agreement within chemical accuracy to full basis diagonalization for molecules like $x\in\mathbb{R}^{2^N}$7 and LiH using only half the qubit count of JW (Shee et al., 2021). Extensions to multi-reference and spin-defect systems, along with hardware-efficient ansätze and error mitigation protocols, achieve CNOT and parameter count reductions of $x\in\mathbb{R}^{2^N}$8–$x\in\mathbb{R}^{2^N}$9 over standard encodings (Huang et al., 2022).
• Hybrid Fermionic–Bosonic Mappings: For systems containing both strongly correlated and predominantly paired electronic regions, splitting spatial orbitals into fully-fermionic and hard-core bosonic (paired) subspaces further reduces the total number of qubits. If $$x \mapsto |\psi\rangle = \sum_{i=0}^{2^N-1} a_i|i\rangle$$0 is the number of fermionic orbitals and $$x \mapsto |\psi\rangle = \sum_{i=0}^{2^N-1} a_i|i\rangle$$1 is the number of bosonic (pairable) orbitals, the total number of qubits becomes $$x \mapsto |\psi\rangle = \sum_{i=0}^{2^N-1} a_i|i\rangle$$2. Circuit depth, CNOT count, and measurement partitioning are all reduced, with Hilbert-space truncation errors well-controlled for appropriately chosen partitions. Adaptive schemes based on orbital occupations and energy gradients optimize this balance (Santos et al., 2024).
• Gray Code Hamiltonian Encodings: Utilizing Gray code mappings to index many-body basis states, one achieves exponential reductions in qubit count and circuit depth compared to one-hot encodings (from $$x \mapsto |\psi\rangle = \sum_{i=0}^{2^N-1} a_i|i\rangle$$3 to $$x \mapsto |\psi\rangle = \sum_{i=0}^{2^N-1} a_i|i\rangle$$4 qubits), and empirical studies demonstrate lower VQE variance and improved noise resilience (Matteo et al., 2020).

4. Specialized Quantum State Preparation Schemes

Efficient state preparation circuits are a key enabler for qubit-efficient encoding. Notable techniques include:

• Superposing Algorithms: Preparation of equal superpositions over arbitrary $$x \mapsto |\psi\rangle = \sum_{i=0}^{2^N-1} a_i|i\rangle$$5 basis states can be achieved in $$x \mapsto |\psi\rangle = \sum_{i=0}^{2^N-1} a_i|i\rangle$$6 qubits with depth and CNOT count scaling linearly, using no ancillas, and achieving a maximum of $$x \mapsto |\psi\rangle = \sum_{i=0}^{2^N-1} a_i|i\rangle$$7 CNOTs—proven optimal for the problem class (Kim et al., 2024).
• Fixed Hamming Weight Encodings: For tasks and Hamiltonians where the relevant quantum state is restricted to a Hamming-weight-$$x \mapsto |\psi\rangle = \sum_{i=0}^{2^N-1} a_i|i\rangle$$8 subspace (e.g., low-lying states of particle-conserving systems or certain quantum machine learning tasks), exact algorithms exist to embed arbitrary $$x \mapsto |\psi\rangle = \sum_{i=0}^{2^N-1} a_i|i\rangle$$9-dimensional data into $T^{s_1\dots s_N}$0 qubits, achieving polynomial compression with $T^{s_1\dots s_N}$1 CNOTs (Farias et al., 2024).
• Local Function Mixture Loading: For real-space quantum chemistry and quantum simulation, deterministic and probabilistic encoding circuits efficiently generate arbitrary linear combinations of $T^{s_1\dots s_N}$2 localized basis functions in $T^{s_1\dots s_N}$3 time and using only $T^{s_1\dots s_N}$4 qubits, controlling amplitude errors via well-defined analytic bounds (Kosugi et al., 2024, Kosugi et al., 13 Jan 2025).

5. Qubit-Efficient Encodings for Quantum Machine Learning

Quantum machine learning faces acute challenges in data-loading and feature representation for high-dimensional classical datasets:

• Qubit-Efficient Amplitude Encoding via Dimensionality Reduction: Quantum Principal Geodesic Analysis (qPGA) projects data onto low-dimensional unit spheres by combining tangent-space PCA with Riemannian exponential maps, reducing the qubit count for amplitude encoding from $T^{s_1\dots s_N}$5 (raw feature space) to $T^{s_1\dots s_N}$6, where $T^{s_1\dots s_N}$7 is the principal subspace dimension capturing prescribed variance. Theoretical bounds for qubit savings and robustness to noise are given, yielding $T^{s_1\dots s_N}$8 classifier accuracy on Quantum SVMs for MNIST/Fashion-MNIST with as few as two qubits. The method is non-invertible, enhancing privacy and attack resistance over autoencoder-based schemes (Cowlessur et al., 24 Jun 2025).
• Quantum Random-Access Coding (QRAC): For discrete features, $T^{s_1\dots s_N}$9-QRACs enable packing $$|\psi_{\rm MPS}\rangle =\sum_{s_1\cdots s_N} \Bigl(\prod_{k=1}^N A^{[k]\,s_k}\Bigr)\, |s_1\cdots s_N\rangle,$$0 classical bits into $$|\psi_{\rm MPS}\rangle =\sum_{s_1\cdots s_N} \Bigl(\prod_{k=1}^N A^{[k]\,s_k}\Bigr)\, |s_1\cdots s_N\rangle,$$1 qubits, such that each bit can be probabilistically recovered with $$|\psi_{\rm MPS}\rangle =\sum_{s_1\cdots s_N} \Bigl(\prod_{k=1}^N A^{[k]\,s_k}\Bigr)\, |s_1\cdots s_N\rangle,$$2. One- and two-qubit QRACs are constructed explicitly to compress categoricals and reduce VQC parameterization, with performance validated on classical datasets. The tradeoff is between qubit savings ($$|\psi_{\rm MPS}\rangle =\sum_{s_1\cdots s_N} \Bigl(\prod_{k=1}^N A^{[k]\,s_k}\Bigr)\, |s_1\cdots s_N\rangle,$$3 by the Nayak bound) and recovery fidelity; as $$|\psi_{\rm MPS}\rangle =\sum_{s_1\cdots s_N} \Bigl(\prod_{k=1}^N A^{[k]\,s_k}\Bigr)\, |s_1\cdots s_N\rangle,$$4 per QRAC increases, classification performance degrades, but overall resource efficiency is markedly improved (Yano et al., 2020).
• Qutrit-Based Encodings: By employing three-level systems, one can encode up to four features per qutrit (compact NCE encoding), achieving high classification accuracy with a reduced number of physical components relative to qubits. Training the data-to-encoding mapping via a dedicated cost function further mitigates sensitivity to feature order and optimizes class separation, as demonstrated on transmon-based hardware (Cao et al., 2023).

6. Space-Efficient Encodings for Bosonic and Many-Body Systems

Bosonic systems and general many-body problems confront a curse of dimensionality in conventional encodings:

• Selective Hamming Truncation: For vibrational and other bosonic simulation tasks, compact or Gray-code encodings are systematically truncated to subspaces of low Hamming weight. This yields flexible trade-offs between qubit count and circuit depth, optimized to the structure of the variational ansatz and NISQ hardware capabilities. Gate count per excitation can be dramatically reduced; the method interpolates between one-hot (lowest depth, highest qubits) and binary (lower qubits, higher depth) mappings (Majland et al., 2021).
• Encoding for Quantum Error Correction: Efficient, measurement-free encoding of logical states for surface codes (e.g., preparing the logical $$|\psi_{\rm MPS}\rangle =\sum_{s_1\cdots s_N} \Bigl(\prod_{k=1}^N A^{[k]\,s_k}\Bigr)\, |s_1\cdots s_N\rangle,$$5 in a distance-3, nine-qubit code) allows fault-tolerant initialization without ancillas in shallow circuits. Extensions via code concatenation to larger distances are possible with moderate overhead and postselected error detection (Goto et al., 2023).

7. Limitations, Open Problems, and Future Directions

All qubit-efficient encoding techniques trade off one or more of circuit depth, ancilla overhead, mapping complexity, or classical preprocessing for qubit savings. Resource scalings can become disadvantageous for highly entangled or unstructured states, or when Hamiltonians are forced to span the full Hilbert space for generality. Measurement partitions and Pauli string weights can increase in compact encodings. For combinatorial optimization encodings that shift constraints into the mixer or circuit structure, efficient implementation of generalized quantum mixers remains a crucial challenge for large instances (Garhofer et al., 2024).

Open directions include:

• Extending MPS and tensor network mapping strategies to broader data modalities and general quantum architectures.
• Systematic benchmarking of hybrid classical-quantum dimensionality reduction (e.g., qPGA) under diverse hardware noise conditions and for privacy-preserving applications.
• Scaling specialized state-preparation circuits for multidimensional and high-cardinality data using ancilla-free or ancilla-efficient methods.
• Developing theory and practice for tunable interpolations between minimal qubit and minimal depth encodings, adaptive to concrete problem and hardware constraints.

Qubit-efficient encoding, when integrated with optimal circuit compilation, hardware-aware design, and advanced error mitigation, forms a cornerstone of contemporary strategy for quantum computational scalability on both NISQ and next-generation devices.

References:

• (Jeon et al., 2024, Garhofer et al., 2024, Kantbekova et al., 9 Sep 2025, Shee et al., 2021, Huang et al., 2022, Santos et al., 2024, Matteo et al., 2020, Kim et al., 2024, Farias et al., 2024, Cowlessur et al., 24 Jun 2025, Yano et al., 2020, Cao et al., 2023, Majland et al., 2021, Goto et al., 2023, Tan et al., 2020, Kosugi et al., 2024, Kosugi et al., 13 Jan 2025)