First-Quantized Qubitization Circuit

• First-quantized qubitization circuits are quantum algorithms that embed Hamiltonians into invariant SU(2) subspaces to enable scalable simulation.
• They employ tailored oracles for state preparation, selective evolution, and reflections to achieve optimal query and gate complexity in simulation tasks.
• Applications in quantum chemistry and condensed matter highlight their resource efficiency for fault-tolerant, large-scale quantum simulation.

A first-quantized qubitization circuit is a quantum algorithmic architecture that implements Hamiltonian simulation in first quantization via qubitization, achieving optimal asymptotic and non-asymptotic query complexity for time evolution and eigenvalue estimation. This circuit structure is characterized by embedding the target Hamiltonian into a larger unitary, such that its action is confined to invariant SU(2) subspaces dictated by the eigenstructure of the Hamiltonian. First-quantized qubitization circuits provide the leading approach for scalable, resource-optimal, and fault-tolerant quantum simulation for a wide class of Hamiltonians, including those relevant for quantum chemistry and condensed matter systems.

1. Mathematical Structure of First-Quantized Qubitization

In first-quantized qubitization, the Hamiltonian $\hat{H}$ acts on a register that explicitly encodes all degrees of freedom (e.g., particle position or momentum indices). The Hamiltonian is written in standard form by decomposing it into oracles: $$\hat{H} = \left(\langle G|\otimes \mathcal{I}\right)\hat{U}\left(|G\rangle\otimes\mathcal{I}\right)$$ where:

• $\hat{U}$ is a unitary oracle (the "block encoding" of $\hat{H}$),
• $|G\rangle$ is an ancilla ("flag" or "prepare") state created by a unitary $\hat{G}$.

When acting on $|G\rangle|\psi\rangle$, $\hat{U}$ yields

$$\hat{U} (|G\rangle|\psi\rangle) = |G\rangle(\hat{H}|\psi\rangle) + |G^\perp\rangle|\psi^\perp\rangle$$

so that postselection (or projection) onto $|G\rangle$ recovers the Hamiltonian action.

A key technical advance is the construction of an "iterate" $\hat{W}$ which, when restricted to two-dimensional invariant subspaces (labeled by eigenvalues $\lambda$ of $\hat{H}$), has block structure

$$\hat{W}_\lambda = \begin{bmatrix} \lambda & -\sqrt{1-\lambda^2}\ \sqrt{1-\lambda^2} & \lambda \end{bmatrix} = \exp(-i Y_\lambda \theta_\lambda),\quad \theta_\lambda = \arccos(\lambda)$$

where $Y_\lambda$ is a Pauli–Y operator in the subspace. This direct sum over all $\lambda$ partitions the Hilbert space into SU(2)-invariant sectors, allowing functions $f(\hat{H})$ to be "programmed" as quantum signal processing sequences of $\hat{W}$ and its controlled powers.

2. Core Algorithmic Ingredients and Oracle Construction

The circuit relies on three unitaries:

• $\hat{G}$: state-preparation, prepares $|G\rangle$,
• $\hat{S}$: reflection about $|G\rangle$ (generated by $2|G\rangle\langle G| - \mathcal{I}$),
• $\hat{U}$: controlled application of the component unitary Hamiltonian terms.

The iterate is then formed as $\hat{W} = \hat{S}'\hat{U}$, with $\hat{S}' = (\hat{R}\otimes\mathcal{I})\hat{S}$ and $\hat{R}$ a reflection over $|G\rangle$.

The necessary and sufficient conditions for this structure to preserve the Hamiltonian embedding are: $$(\langle G| \otimes \mathcal{I}) \hat{S} \hat{U} (|G\rangle \otimes \mathcal{I}) = \hat{H},\qquad (\langle G| \otimes \mathcal{I}) \hat{S} \hat{U} \hat{S} \hat{U} (|G\rangle \otimes \mathcal{I}) = \mathcal{I}$$ This structure ensures that repeated application of $\hat{W}$ evolves only within SU(2) sectors, enabling the use of quantum signal processing for arbitrary analytic functions of $\hat{H}$.

In typical first-quantized simulations (such as electrons represented in a plane-wave or real-space basis), the Hamiltonian is decomposed into a Linear Combination of Unitaries (LCU)

$H = \sum_\ell \alpha_\ell H_\ell$

A PREPARE oracle creates $\sum_\ell \sqrt{\alpha_\ell/\lambda}|\ell\rangle$; a SELECT oracle applies $H_\ell$ conditioned on $|\ell\rangle$. The block-encoding then satisfies $$\langle 0|\mathrm{PREP}^\dagger\, \mathrm{SEL}\, \mathrm{PREP}|0\rangle = H/\lambda$$, where $\lambda = \sum_\ell \alpha_\ell$ (Su et al., 2021).

3. Circuit Architecture, Ancilla Usage, and Gate Complexity

Optimality is achieved in both asymptotic and practical regimes:

• Query complexity: $Q = \mathcal{O}(t + \log(1/\epsilon))$ for simulating $e^{-i\hat{H}t}$ to error $\epsilon$, matching lower bounds across all regimes (Low et al., 2016).
• Ancilla Usage: At most two additional ancilla qubits (beyond those for $\hat{G}$ and $\hat{U}$) are required, outperforming prior art (e.g., $d$-sparse Hamiltonian simulation) which needs linear ancilla overhead (Low et al., 2016).
• Gate Complexity: For an ancilla sub-register of dimension $d$, only $\mathcal{O}(Q\log d)$ additional 2-qubit gates are needed.

A key architectural feature is the use of controlled versions of $\hat{G}$, $\hat{U}$, and often their inverses (e.g., controlled swaps, controlled phase rotations, oracles for arithmetic or bitwise multiplication), with all non-trivial action confined to small, well-structured subspaces.

Additionally, improvements in block-encoding and arithmetic routines—for example, applying Hamiltonian coefficients via QROM in parallel, or replacing Toffoli-encoded control with sequences of Pauli string rotations and gadget circuits—further lower overheads, especially when using unary or generalized unary encoding (Steudtner et al., 2019). In these schemes, parallelization and/or tree-based data structures can achieve circuit depth scaling as low as $\mathcal{O}(\log \Lambda)$ for Hamiltonians with $\Lambda$ terms, trading ancilla qubits for depth.

4. Applications, Simulation Scaling, and Fault-Tolerant Resource Estimates

First-quantized qubitization circuits have been established as the most resource-efficient protocol for simulating large molecular systems and materials in the fault-tolerant setting:

• Scaling: For a system of $N$ electrons in $M$ orbitals (plane-wave basis), the gate complexity for QPE is

$$\tilde{\mathcal{O}}\left(\frac{N^{4/3}M^{2/3} + N^{8/3}M^{1/3}}{\varepsilon}\right)$$

where $\tilde{\mathcal{O}}$ hides polylogarithmic factors and $\varepsilon$ is the desired absolute precision (Ku et al., 2 Oct 2025, Su et al., 2021).

• Qubit requirements: $\mathcal{O}(N\log M)$, compared to $\mathcal{O}(M)$ in second quantization.

The strict linear dependence on $1/\varepsilon$ (in contrast to higher order in Trotterization or product formula methods), and sublinear dependence on $M$, enables quantum simulation at chemical accuracy for systems with millions of plane-wave orbitals, at gate counts many orders of magnitude lower than previously possible.

For example, resource estimates for fault-tolerant simulation of realistic molecules (tens of electrons, thousands of plane waves) project T gate counts and spacetime volumina far lower than any known second-quantized or Trotterized approach (Ku et al., 2 Oct 2025). These circuits are thus expected to be preferred in large-scale, fault-tolerant quantum chemistry and condensed matter simulation.

5. Variants, Hybridization, and Generalizations

Variants of the first-quantized qubitization circuit have been developed:

• Interaction-picture algorithms: Achieve comparable asymptotic complexity ($\mathcal{O}(\eta^{8/3}N^{1/3}t)$), but with larger constant overheads due to additional time register sorting/amplitude amplification.
• Divide-and-conquer block-encoding: For tasks such as light–matter interaction (Pauli–Fierz Hamiltonians), recursively grouping Hamiltonian fragments and employing advanced block-encoding synthesis further optimize simulation cost (Mukhopadhyay et al., 2023).
• Hybrid quantization schemes: Conversion circuits between first- and second-quantized encodings (at gate cost $\mathcal{O}(N\log N\log M)$) enable efficient state preparation and measurement, combining the advantages of both representations in composite simulations (Ku et al., 6 Jul 2025).

Extensions of first-quantized qubitization to specialized scenarios—such as systems with external fields (Kosugi et al., 2022), non-trivial particle statistics (Toppan, 2022), or structured oscillator networks (Lee et al., 2023)—demonstrate the broad generality and adaptability of the approach.

6. Practical Limitations and Implementation Considerations

Despite optimal asymptotic scaling, practical deployment on hardware requires attention to several subtleties:

• Efficient Oracle Realization: Realizing oracles for state preparation (PREPARE) and term selection (SELECT) dominates the circuit cost. Optimized schemes exploit quantum read-only memory (QROM), sparse arithmetic, and recycling of ancilla registers (Su et al., 2021).
• Ancilla and Connectivity Tradeoffs: Unary or tree-based encodings remove Toffoli depth but increase the number of ancilla qubits and impose non-trivial connectivity constraints, potentially introducing swaps or reducing parallelism on hardware with limited qubit interconnectivity (Steudtner et al., 2019).
• Sparsity and Data Locality: The efficacy of block-encoding and simulation cost depends critically on the structure and sparsity of the Hamiltonian. For well-structured/sparse systems (e.g., many solid-state and quantum chemistry models), the first-quantized circuit is especially effective.
• Resilience to Decoherence: The depth of the qubitization-based circuit is minimized relative to Trotter approaches, which translates directly into reduced decoherence on fault-tolerant quantum hardware.
• Hybrid and Adaptive Algorithms: Practical quantum chemistry simulation may interleave first-quantized time evolution with measurement protocols and state transformations in second quantization, with conversion circuits amortizing their cost when $N \ll M$ (Ku et al., 6 Jul 2025).

7. Summary Table: Core Resource Metrics

Approach	Qubit Count	Gate Scaling (to error ε)	Distinctive Feature
First-Quantized Qubitization	$\mathcal{O}(N\log M)$	$$\tilde{\mathcal{O}}\big((N^{4/3}M^{2/3} + N^{8/3}M^{1/3})/\varepsilon\big)$$	Optimal in both $M$ and $1/\varepsilon$
Second-Quantized Trotterization	$\mathcal{O}(M)$	$\mathcal{O}(M^7/\varepsilon^2)$	Higher gate count, $M\gg N$
First-Quantized Interaction Picture	$\mathcal{O}(N\log M)$	$\tilde{\mathcal{O}}(N^{8/3}M^{1/3}/\varepsilon)$	Asymptotically competitive
Divide-and-Conquer/Trotterization	System-dependent	Varies (can outperform qubitization for high-cutoff QED problems)	Adaptive to Hamiltonian structure

Each resource figure traces directly to explicit data and equations in the relevant literature (Low et al., 2016, Su et al., 2021, Ku et al., 2 Oct 2025, Mukhopadhyay et al., 2023, Steudtner et al., 2019).

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First-quantized qubitization circuits are now established as the leading solution for high-precision, large-scale quantum simulation in chemistry, materials, and beyond, owing to their unique combination of optimal query complexity, low ancilla overhead, adaptability to hybrid schemes, and direct hardware relevance in the fault-tolerant regime.