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arXiv:2608.16937

New Methods and Frameworks for Quantum State Preparation in Modern Quantum Systems

Alexis Gomez

quant-phSubmitted Aug 10, 2026• Announced Aug 11, 2026• 0 citations
arXivHTMLPDF

Abstract

This thesis studies exact, deterministic preparation of arbitrary dense n-qubit states, the data-loading step in quantum signal and image processing. It derives two syntheses built on the Digital Signal-induced Heap Transform (DsiHT): the QsiHT Fast Path Real Synthesis and the QsiHT Fast Path Complex Synthesis. Both are benchmarked against ten configurations spanning the UCR, isometry, multiplexor, Schmidt/SVD, QSD, and heap-transform families. Several of those are realizations through Qiskit builders or compiler optimization, not from-scratch reimplementations. The n=3 noisy comparison spans ibm_fez, ibm_kingston, and ibm_marrakesh, three 156-qubit IBM Heron r2 processors, where single-submission nine-method jobs permit within-session family-wide Benjamini-Hochberg-corrected comparisons. The frontier-versus-QSD separation reproduces within one calibration on every device, but the within-frontier order does not reproduce across devices or calibration days. A deep-circuit n=8 run on ibm_fez shows the executed-count ordering re-emerge outside the run-to-run spread on the complex target. Depth was not isolated from mapping, gate composition, calibration, or session effects. All methods are exact to machine precision and differ only in cost. Under noise the coarse error tier tracks the executed (routed) two-qubit count, separating the Theta(2^n) frontier from QSD's Theta(4^n) and nothing finer. Both syntheses realize the deployed Qiskit StatePreparation floor of 2^n-n-1 CNOTs, undercutting every other from-scratch method on the as-built CNOT axis. The Real Synthesis serves real (sign-bearing) targets and holds the lowest classical build cost among the exact loaders at large register sizes, through one fast Walsh-Hadamard pass. The Complex Synthesis serves arbitrary complex targets and ties Qiskit's StatePreparation for the lowest simulated sampled error.

Related papers

New effective methods of preparation of states of quantum image representation

Artyom M. Grigoryan, Alexis A. Gomez, Sos С. Agaian

The multi-qubit transformation is the important step in quantum computation in many applications. In recent works, this problem is addressed by solving by using the QR decomposition with elementary rotations with one- and two-qubit CNOT gates. Reducing the number of these gates is critical since operations on three or more qubits are difficult to implement on the current quantum computers. The quantum signal induced heap transform (QsiHT) provides an efficient method for state preparation in the realm of quantum image representation by amplitudes. This paper presents a detailed analysis of how the QsiHT, particularly through the strong path, is utilized to construct quantum states to represent images. The algorithm behind this method and the realization of the corresponding quantum circuits is outlined. Practical examples of the QsiHT on 2 and 3-qubit systems are given using the Qiskit Framework for quantum circuit simulation.

Multimodal Image Exploitation and Learning•May 28, 2025•0 citations

A Novel Approach to State-to-State Transformation in Quantum Computing

Artyom M. Grigoryan, Alexis A. Gomez, Sos С. Agaian

The article presents a new approach to the problem of transforming one quantum state into another. It is shown that a r-qubit superposition |x can be obtained from another r-qubit superposition |y, by using only (2r-1) rotations, each presented by one controlled rotation gate. The quantum superpositions with real amplitudes are considered. The traditional two-stage approach Uy-1Ux:|x→|0⊕r→|y requires twice as many rotations. Here, both transformations to the conventual basis state, Ux: |x→ |0⊕r and Uy: |y→ |0⊕r, use (2r-1) rotations each on two binary planes and many of these rotations require additional sets of CNOTs to be represented as 1- or 2-qubit controlled gates. The proposed method is based on the concept of the discrete signal-induced heap transform (DsiHT) which is unitary and generated by a vector and a set of angular equations with given parameters. The quantum analogue of this transform is described. The main characteristic of the DsiHT is the path of processing the data. It is shown that exist such fast paths that allow for effective computing of the DsiHT, which leads to the simple quantum circuits for state preparation and transformation. Examples of such paths are given and quantum circuits for preparation and transformation of 2-, 3-, and 4-qubits are described in detail. CNOT gates are not used, but only controlled gates of elementary rotations around the y-axis. It is shown that the transformation and, in particular, only rotation gates with control qubits are required for initialization of 4-qubits. The quantum circuits are simple and have a recursive form, which makes them easy to implement for arbitrary r-qubit superposition, with r≥2. This approach significantly reduces the complexity of quantum state transformations, paving the way for more efficient quantum algorithms and practical implementations on near-term quantum devices.

Information•Aug 13, 2025•5 citations

Ground-State Preparation by Projection onto the Maximal Decoherence-Free Subspace: Operator-Algebraic Derivation and Constant-Depth Execution on 156-Qubit Processors

Mohamed Hassan

Abstract We present a framework for preparing quantum ground states by structural projection rather than by deep-circuit Trotterized dynamical simulation. The method is grounded in a rigorous operator-algebraic formalization spanning 297 parts (the IQS/EQS corpus), from which we extract the complete derivation chain 𝐻_boundary → 𝐴_𝜇 → 𝑃_𝐻 → 𝐴_𝐻. We prove that the Host projector 𝑃_𝐻 = 𝑃_DFS^max acts as a native state-preparation oracle — a single projective step applied once, not a query oracle in the phase-kickback sense: assuming appropriate overlap, it projects any initial state |𝜓₀⟩ into the prescribed ground-state subspace with an 𝒪(1) execution depth on the quantum device.The body of this paper is confined to the framework and its theorems; the complete experimental record is given in the appendix. We validate the framework through 55 independent QPU executions on three IBM Heron-class processors (156 qubits, 𝑟2 revision): ibm_marrakesh, ibm_kingston and ibm_fez. Twenty-one of these form the Phase 3–6 programme, utilizing up to 76 actively measured qubits and achieving highly non-uniform output distributions in which the most probable outcome exceeds its uniform-distribution probability by factors of 3.44 × 10⁵ to 5.50 × 10⁶ at 50,000 shots; the remaining twenty-four test derived theorems directly against predictions computed and published before submission, including the oracle projection at 𝑁 = 6 to 12, where up to 82.1% of shots return an exact maximum cut of a frustrated graph against a uniform null of 4.88 × 10⁻⁴. We demonstrate cross-domain applicability by deploying custom Hamiltonians encoding molecular chemistry (H₂ electron correlation), integer factorization (cryptographic Ising reduction), and combinatorial optimization (MaxCut/portfolio) into the same oracle architecture, all executing at identical 𝒪(1) quantum circuit depth. We further extend the oracle to 𝑘-local Hamiltonians (𝑘 = 3, 4, 5) utilizing up to 153 actively measured qubits per execution, proving via conservation of the total magnetization 𝐴₁ = ∑ₓ 𝑍ₓ that the framework generalizes to a specific structured family of 𝑘-body interactions. Finally, we address the problem encoding question by formalizing the Compression Encoding Theorem, which provides the algebraic structure to map target problem families into the oracle (with classically-bounded construction complexity). We instantiate this construction across 6 representative encoded problem instances on live hardware. We are explicit about what this does not claim. The 𝒪(1) figure is the coherent execution depth alone, and the hardness of a target problem is confined rather than removed: to the classical synthesis of the Host isometry in the deterministic route, and to post-selection overhead in the measurement-conditioned one. The encoding theorem bounds that classical cost only on a class whose ground-state energy is computable in polynomial time, and a family whose ground-state energy is NP-hard to compute does not belong to it.These results establish that the quantum execution phase of ground-state preparation is achievable on current NISQ devices without variational optimization, Trotterization, or quantum error correction, provided the problem Hamiltonian is mapped through the decoherence-free-subspace projection framework defined herein.

•Sep 9, 2026•0 citations

QFlux: Quantum Circuit Implementations of Molecular Dynamics. Part III - State Initialization and Unitary Decomposition

Alexander V. Soudackov, Delmar G. A. Cabral, Brandon Allen +7 more

This tutorial builds upon the foundations established in Part~II to present a unified, implementation-oriented overview of quantum state initialization and unitary decomposition for n -qubit systems, available in QFlux . We begin with the preparation of arbitrary quantum states and develop two complementary constructions: (i) a recursive multiplexor method (after Shende et al.) that disentangles qubits via multiplexed R y and R z rotations, and (ii) an algebraic scheme based on uniformly controlled rotations (UCRs) (after Möttönen et al.) that realizes the same mapping through analytically defined rotation networks with predictable gate counts. We then extend these state-preparation tools to generic unitary synthesis through Givens-rotation, column-by-column, and recursive cosine-sine (CSD) and quantum Shannon (QSD) decompositions, explicitly linking linear-algebraic factorizations to executable circuits while tracking CNOT complexity. Finally, we introduce the Walsh decomposition for diagonal unitaries and show how Gray-code ordering and local CNOT cancellations yield an O(2^n) entangling-gate cost, providing shallow, NISQ-friendly implementations. Taken together, these techniques form a pedagogical bridge from matrix analysis to hardware-efficient circuit constructions, offering clear design rules, closed-form parameters, and scalable synthesis pathways for simulation and experiment.

•Jan 30, 2026•0 citations

New Permutation-Free Quantum Circuits for Implementing 3- and 4-Qubit Unitary Operations

Artyom M. Grigoryan

The article presents the quantum signal-induced heap transform (QsiHT) method of the QR-decomposition of multi-qubit operations. This transform can be generated by a given signal, by using different paths, or orders, of processing the data. We propose using the concept of the fast path of calculation of the QsiHT and applying such transforms on each stage of the matrix decomposition. This allows us to build quantum circuits for multi-qubit unitary operation without permutations. Unitary operations with real and complex matrices are considered. The cases of 3- and 4-qubit operations are described in detail with quantum circuits. These circuits use a maximum of 28 and 120 Givens rotation gates for 3- and 4-qubit real operations, respectively. All rotations are performing only on adjacent bit planes. For complex unitary operation, each of the Givens gates is used in pairs with two Z-rotation gates. These two types of rotations and the global phase gate are the universal gate set for multi-qubit operations. The presented approach can be used for implementing quantum circuits for n-qubits when n≥2, with a maximum of (4n/2−2n−1) Givens rotations and no permutations.

Information•Jul 21, 2025•4 citations

Unitary Operations with Five and More Qubits: Roadmaps and Effective Quantum Circuits

Artyom M. Grigoryan

This article presents the general method of QR decomposition of r-qubit operations, r≥ 3, by means of quantum signal-induced heap transformations (QsiHT). These are quantum analogues of discrete signal-induced heap transformations, which are generated by given signals and paths, or orders, of processing the data. The case of the 5-qubit operations is described in detail, and a recurrent form of calculation of all 5-qubit QsiHTs from the 4-qubit QsiHTs is given. For that, roadmaps and quantum circuits are presented for all 31 5-point QsiHTs that are used in the QR decomposition. New roadmaps, namely the schemes with paths for performing basic operations on qubits, and corresponding quantum circuits, are also described for the 4- and 3-qubit transformations. All QsiHTs use fast paths, which allow us to calculate the QR decomposition only on disjoint bit planes. As a result, we build the quantum circuits for 5- and more-qubit operations without permutations, only elementary rotations. Unitary operations with real numbers are considered. In the general case, the method of compositing all the roadmaps and quantum schemes for the calculation of any r-qubit operation by only (2r−1)2r−1 elementary rotations is described.

Information•Feb 6, 2026•0 citations
This thesis studies exact, deterministic preparation of arbitrary dense n-qubit states, the data-loading step in quantum signal and image processing. It derives two syntheses built on the Digital Signal-induced Heap Transform (DsiHT): the QsiHT Fast Path Real Synthesis and the QsiHT Fast Path Complex Synthesis. Both are benchmarked against ten configurations spanning the UCR, isometry, multiplexor, Schmidt/SVD, QSD, and heap-transform families. Several of those are realizations through Qiskit builders or compiler optimization, not from-scratch reimplementations.
The n=3 noisy comparison spans ibm_fez, ibm_kingston, and ibm_marrakesh, three 156-qubit IBM Heron r2 processors, where single-submission nine-method jobs permit within-session family-wide Benjamini-Hochberg-corrected comparisons. The frontier-versus-QSD separation reproduces within one calibration on every device, but the within-frontier order does not reproduce across devices or calibration days. A deep-circuit n=8 run on ibm_fez shows the executed-count ordering re-emerge outside the run-to-run spread on the complex target. Depth was not isolated from mapping, gate composition, calibration, or session effects.
All methods are exact to machine precision and differ only in cost. Under noise the coarse error tier tracks the executed (routed) two-qubit count, separating the Theta(2^n) frontier from QSD's Theta(4^n) and nothing finer. Both syntheses realize the deployed Qiskit StatePreparation floor of 2^n-n-1 CNOTs, undercutting every other from-scratch method on the as-built CNOT axis. The Real Synthesis serves real (sign-bearing) targets and holds the lowest classical build cost among the exact loaders at large register sizes, through one fast Walsh-Hadamard pass. The Complex Synthesis serves arbitrary complex targets and ties Qiskit's StatePreparation for the lowest simulated sampled error.
The multi-qubit transformation is the important step in quantum computation in many applications. In recent works, this problem is addressed by solving by using the QR decomposition with elementary rotations with one- and two-qubit CNOT gates. Reducing the number of these gates is critical since operations on three or more qubits are difficult to implement on the current quantum computers. The quantum signal induced heap transform (QsiHT) provides an efficient method for state preparation in the realm of quantum image representation by amplitudes. This paper presents a detailed analysis of how the QsiHT, particularly through the strong path, is utilized to construct quantum states to represent images. The algorithm behind this method and the realization of the corresponding quantum circuits is outlined. Practical examples of the QsiHT on 2 and 3-qubit systems are given using the Qiskit Framework for quantum circuit simulation.
The article presents a new approach to the problem of transforming one quantum state into another. It is shown that a r-qubit superposition |x can be obtained from another r-qubit superposition |y, by using only (2r-1) rotations, each presented by one controlled rotation gate. The quantum superpositions with real amplitudes are considered. The traditional two-stage approach Uy-1Ux:|x→|0⊕r→|y requires twice as many rotations. Here, both transformations to the conventual basis state, Ux: |x→ |0⊕r and Uy: |y→ |0⊕r, use (2r-1) rotations each on two binary planes and many of these rotations require additional sets of CNOTs to be represented as 1- or 2-qubit controlled gates. The proposed method is based on the concept of the discrete signal-induced heap transform (DsiHT) which is unitary and generated by a vector and a set of angular equations with given parameters. The quantum analogue of this transform is described. The main characteristic of the DsiHT is the path of processing the data. It is shown that exist such fast paths that allow for effective computing of the DsiHT, which leads to the simple quantum circuits for state preparation and transformation. Examples of such paths are given and quantum circuits for preparation and transformation of 2-, 3-, and 4-qubits are described in detail. CNOT gates are not used, but only controlled gates of elementary rotations around the y-axis. It is shown that the transformation and, in particular, only rotation gates with control qubits are required for initialization of 4-qubits. The quantum circuits are simple and have a recursive form, which makes them easy to implement for arbitrary r-qubit superposition, with r≥2. This approach significantly reduces the complexity of quantum state transformations, paving the way for more efficient quantum algorithms and practical implementations on near-term quantum devices.
Abstract We present a framework for preparing quantum ground states by structural projection rather than by deep-circuit Trotterized dynamical simulation. The method is grounded in a rigorous operator-algebraic formalization spanning 297 parts (the IQS/EQS corpus), from which we extract the complete derivation chain 𝐻_boundary → 𝐴_𝜇 → 𝑃_𝐻 → 𝐴_𝐻. We prove that the Host projector 𝑃_𝐻 = 𝑃_DFS^max acts as a native state-preparation oracle — a single projective step applied once, not a query oracle in the phase-kickback sense: assuming appropriate overlap, it projects any initial state |𝜓₀⟩ into the prescribed ground-state subspace with an 𝒪(1) execution depth on the quantum device.The body of this paper is confined to the framework and its theorems; the complete experimental record is given in the appendix. We validate the framework through 55 independent QPU executions on three IBM Heron-class processors (156 qubits, 𝑟2 revision): ibm_marrakesh, ibm_kingston and ibm_fez. Twenty-one of these form the Phase 3–6 programme, utilizing up to 76 actively measured qubits and achieving highly non-uniform output distributions in which the most probable outcome exceeds its uniform-distribution probability by factors of 3.44 × 10⁵ to 5.50 × 10⁶ at 50,000 shots; the remaining twenty-four test derived theorems directly against predictions computed and published before submission, including the oracle projection at 𝑁 = 6 to 12, where up to 82.1% of shots return an exact maximum cut of a frustrated graph against a uniform null of 4.88 × 10⁻⁴. We demonstrate cross-domain applicability by deploying custom Hamiltonians encoding molecular chemistry (H₂ electron correlation), integer factorization (cryptographic Ising reduction), and combinatorial optimization (MaxCut/portfolio) into the same oracle architecture, all executing at identical 𝒪(1) quantum circuit depth. We further extend the oracle to 𝑘-local Hamiltonians (𝑘 = 3, 4, 5) utilizing up to 153 actively measured qubits per execution, proving via conservation of the total magnetization 𝐴₁ = ∑ₓ 𝑍ₓ that the framework generalizes to a specific structured family of 𝑘-body interactions. Finally, we address the problem encoding question by formalizing the Compression Encoding Theorem, which provides the algebraic structure to map target problem families into the oracle (with classically-bounded construction complexity). We instantiate this construction across 6 representative encoded problem instances on live hardware. We are explicit about what this does not claim. The 𝒪(1) figure is the coherent execution depth alone, and the hardness of a target problem is confined rather than removed: to the classical synthesis of the Host isometry in the deterministic route, and to post-selection overhead in the measurement-conditioned one. The encoding theorem bounds that classical cost only on a class whose ground-state energy is computable in polynomial time, and a family whose ground-state energy is NP-hard to compute does not belong to it.These results establish that the quantum execution phase of ground-state preparation is achievable on current NISQ devices without variational optimization, Trotterization, or quantum error correction, provided the problem Hamiltonian is mapped through the decoherence-free-subspace projection framework defined herein.
This tutorial builds upon the foundations established in Part~II to present a unified, implementation-oriented overview of quantum state initialization and unitary decomposition for n -qubit systems, available in QFlux . We begin with the preparation of arbitrary quantum states and develop two complementary constructions: (i) a recursive multiplexor method (after Shende et al.) that disentangles qubits via multiplexed R y and R z rotations, and (ii) an algebraic scheme based on uniformly controlled rotations (UCRs) (after Möttönen et al.) that realizes the same mapping through analytically defined rotation networks with predictable gate counts. We then extend these state-preparation tools to generic unitary synthesis through Givens-rotation, column-by-column, and recursive cosine-sine (CSD) and quantum Shannon (QSD) decompositions, explicitly linking linear-algebraic factorizations to executable circuits while tracking CNOT complexity. Finally, we introduce the Walsh decomposition for diagonal unitaries and show how Gray-code ordering and local CNOT cancellations yield an O(2^n) entangling-gate cost, providing shallow, NISQ-friendly implementations. Taken together, these techniques form a pedagogical bridge from matrix analysis to hardware-efficient circuit constructions, offering clear design rules, closed-form parameters, and scalable synthesis pathways for simulation and experiment.
The article presents the quantum signal-induced heap transform (QsiHT) method of the QR-decomposition of multi-qubit operations. This transform can be generated by a given signal, by using different paths, or orders, of processing the data. We propose using the concept of the fast path of calculation of the QsiHT and applying such transforms on each stage of the matrix decomposition. This allows us to build quantum circuits for multi-qubit unitary operation without permutations. Unitary operations with real and complex matrices are considered. The cases of 3- and 4-qubit operations are described in detail with quantum circuits. These circuits use a maximum of 28 and 120 Givens rotation gates for 3- and 4-qubit real operations, respectively. All rotations are performing only on adjacent bit planes. For complex unitary operation, each of the Givens gates is used in pairs with two Z-rotation gates. These two types of rotations and the global phase gate are the universal gate set for multi-qubit operations. The presented approach can be used for implementing quantum circuits for n-qubits when n≥2, with a maximum of (4n/2−2n−1) Givens rotations and no permutations.
This article presents the general method of QR decomposition of r-qubit operations, r≥ 3, by means of quantum signal-induced heap transformations (QsiHT). These are quantum analogues of discrete signal-induced heap transformations, which are generated by given signals and paths, or orders, of processing the data. The case of the 5-qubit operations is described in detail, and a recurrent form of calculation of all 5-qubit QsiHTs from the 4-qubit QsiHTs is given. For that, roadmaps and quantum circuits are presented for all 31 5-point QsiHTs that are used in the QR decomposition. New roadmaps, namely the schemes with paths for performing basic operations on qubits, and corresponding quantum circuits, are also described for the 4- and 3-qubit transformations. All QsiHTs use fast paths, which allow us to calculate the QR decomposition only on disjoint bit planes. As a result, we build the quantum circuits for 5- and more-qubit operations without permutations, only elementary rotations. Unitary operations with real numbers are considered. In the general case, the method of compositing all the roadmaps and quantum schemes for the calculation of any r-qubit operation by only (2r−1)2r−1 elementary rotations is described.