This paper presents a quantum probability splitter, i.e. a quantum circuit that modifies the probability amplitudes of a qubit so that the probability on the selected basis state is halved. It also presents a potential application of this circuit related to qubit preparation: it shows how a qubit is prepared to a superposition of any valid pair of basis states probabilities, by repeated application of the quantum probability splitter.
This paper presents a quantum probability splitter, i.e. a quantum circuit that modifies the probability amplitudes of a qubit so that the probability on the selected basis state is halved. It also presents a potential application of this circuit related to qubit preparation: it shows how a qubit is prepared to a superposition of any valid pair of basis states probabilities, by repeated application of the quantum probability splitter.
An approximate quantum state preparation method is introduced, called the Walsh series loader (WSL). The WSL approximates quantum states defined by real-value functions of single real variables with a depth independent of the number n of qubits. Two approaches are presented. The first one approximates the target quantum state by a Walsh series truncated at $O(1/\sqrt{\ensuremath{\epsilon}})$, where $\ensuremath{\epsilon}$ is the precision of the approximation in terms of infidelity. The circuit depth is also $O(1/\sqrt{\ensuremath{\epsilon}})$, the size is $O(n+1/\sqrt{\ensuremath{\epsilon}})$, and only one ancilla qubit is needed. The second method accurately represents quantum states with sparse Walsh series. The WSL loads s-sparse Walsh series into n qubits with a depth doubly sparse in s and k, the maximum number of bits with value 1 in the binary decomposition of the Walsh function indices. The associated quantum circuit approximates the sparse Walsh series up to an error $\ensuremath{\epsilon}$ with a depth O(sk), a size O(n+sk), and one ancilla qubit. In both cases, the protocol is a repeat-until-success procedure with a probability of success $P=\mathrm{\ensuremath{\Theta}}(\ensuremath{\epsilon})$, giving an averaged total time of $O(1/{\ensuremath{\epsilon}}^{3/2})$ for the WSL and $O(sk/\ensuremath{\epsilon})$ for the sparse WSL. Amplitude amplification can be used to reach a probability of success $P=\mathrm{\ensuremath{\Theta}}(1)$, modifying the quantum circuit size to $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}((n+1/\sqrt{\ensuremath{\epsilon}})/\sqrt{\ensuremath{\epsilon}})$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}((n+sk)/\sqrt{\ensuremath{\epsilon}})$ and the depth to $O([log{(n)}^{3}+1/\sqrt{\ensuremath{\epsilon}}]/\sqrt{\ensuremath{\epsilon}})$ and $O([log{(n)}^{3}+sk]/\sqrt{\ensuremath{\epsilon}})$, respectively. Amplitude amplification reduces by a factor $O(1/\sqrt{\ensuremath{\epsilon}})$ the total time dependence on $\ensuremath{\epsilon}$ but increases the size and depth of the associated quantum circuits, making them linearly dependent on n. These protocols give overall efficient algorithms with no exponential scaling in any parameter. They can be generalized to any complex-value, multivariate, almost-everywhere-differentiable function. The repeat-until-success Walsh series loader is so far the only method that prepares a quantum state with a circuit depth and an averaged total time independent of the number of qubits.
The preparation of a quantum superposition is the key to the success of many quantum algorithms and quantum machine learning techniques. The preparation of an incomplete or a non-uniform quantum superposition with certain properties is a non-trivial task. In this paper, ann-qubits variational quantum circuit using partial negation and controlled partial negation operators is proposed to prepare a quantum superposition from a given space of probability distributions. The speed of the preparation process and the accuracy of the prepared superposition has special importance to the success of any quantum algorithm. The proposed method can be used to prepare the required quantum superposition inO(n)steps and with high accuracy when compared with relevant methods in literature.
This record contains the Villaber Quantum Educational Poster Series v1.0 - FINAL COMPLETE CERTIFIED 2026, a collection of 6 educational scientific posters that visually explain the fundamentals of quantum computing. ABSTRACT: This educational series provides a complete, step-by-step visual pathway from single-qubit superposition to two-qubit entanglement. It is designed as an Open Educational Resource (OER) for students, educators, and researchers in Quantum Computing and Quantum Information Science. CONTENTS OF THE COLLECTION: 1. First Step in Quantum Computing: Understanding the Qubit - Bloch Sphere visualization of qubit state |ψ⟩ = (1/√2)|0⟩ + (1/√2)|1⟩ - Proof of total probability: P0 = |1/√2|² = 0.5, P1 = |1/√2|² = 0.5, Ptotal = 1.0 = 100% - Classical Bit (0 or 1) vs. Qubit (both 0 and 1, like a spinning coin) - Importance of superposition for massive parallel processing. 2. Step 2 in Quantum Computing: Entanglement & CNOT Gate - What is Entanglement: inseparable states where measuring one qubit instantly determines the other. - Bell State Formula: |Φ+⟩ = (1/√2)|00⟩ + (1/√2)|11⟩ with 50% |00⟩ and 50% |11⟩ outcome. - How to Create Entanglement: 2-qubit circuit with Hadamard (H) gate on q0 and CNOT gate with q0 as control and q1 as target. - CNOT Truth Table and why it matters for quantum algorithms, teleportation, and error correction. 3. How to Make Bell State Using Qubit 1 & Qubit 2 - Villaber Quantum - STEP 0 - INITIAL STATE: |ψ0⟩ = |0⟩ ⊗ |0⟩ = |00⟩ - STEP 1 - APPLY HADAMARD GATE TO QUBIT 1: Qubit 1 placed into superposition |ψ1⟩ = 1/√2|00⟩ + 1/√2|10⟩ - STEP 2 - APPLY CNOT GATE: If q0=|1⟩ then flip q1, resulting state |ψ2⟩ = 1/√2|00⟩ + 1/√2|11⟩ - STEP 3 - FINAL BELL STATE |Φ+⟩: |Φ+⟩ = 1/√2|00⟩ + 1/√2|11⟩, maximally entangled with 50/50 probability. Includes quantum circuit diagram: H gate on q0, then CNOT with control q0 and target q1. 4. Final Complete Qubit 1 and 2 Passed - Universal Halving Principle - Qubit 1 Superposition verification: State |ψ⟩ = (1/√2)|0⟩ + (1/√2)|1⟩, Total = 0.5 + 0.5 = 1, Normalized - Verified Correct. - Two-Qubit Superposition verification: State |ψ⟩ = (1/2)|00⟩ + (1/2)|01⟩ + (1/2)|10⟩ + (1/2)|11⟩, Probabilities 4 × (1/2)² = 1. - Universal Halving Principle - Infinite Series: Σ (1/2^n) = 1 from n=1 to ∞, proving conservation = probability one. - Assessment Checklist PASSED: Fraction, Algebra, Geometry, Probability, Quantum Normalization, Entanglement, H/CNOT Gates, Measurement. LEARNING GOAL: Create maximally entangled Bell pair |Φ+⟩ and demonstrate Bell State preparation using standard gate operations. KEY TAKEAWAY: Entanglement = Correlated States, CNOT = Core 2-Qubit Gate for Entanglement. Superposition is what gives quantum computers the power to explore many possible solutions at once. ISSUED: 2026 | VILLABER LABS QUANTUM CERTIFICATION DIVISION | Certificate ID: VQF-FC-Q12-2026-100P STATUS: FINAL COMPLETE - PASSED - VALID 2026 This work is licensed under CC BY 4.0.
As interest in quantum computing evolves, consideration must be given to the development of new methods to improve the current design of quantum computers. Such ideas are not only helping the advance towards practical quantum computation applications, but are also providing clearer understanding of quantum computation itself. Eventually, several new exploratory efforts to increase the efficiency beyond the inherent advantage of quantum computational systems to classical systems will materialize. As a part of these exploration efforts, this paper presents a modified version of the qubit, which we refer to as a "Qubit", that allows a smaller number of Qubits than qubits to reach the same result in applications such as Shor’s algorithm for the factorization of large numbers. The current model of the qubit consists of a quantum bit with two states, a zero and a one in a quantum superposition state. The Qubit, which consists of more than two states, is introduced and explained. A mathematical analysis of the Qubit within Hilbert space is given. We present examples of applications of the Qubit to several quantum computing algorithms, including discussion of the advantages and disadvantages that are involved. Finally a physical model to construct such a Qubit is considered.
We present short-depth quantum circuits to deterministically prepare any Dicke state $\left| {D_k^n} \right\rangle $, which is the equal-amplitude superposition of all n-qubit computational basis states with Hamming Weight k. Dicke states are an important class of entangled quantum states with a large variety of applications, and a long history of experimental creation in physical systems. On the other hand, not much is known regarding efficient scalable quantum circuits for Dicke state preparation on realistic quantum computing hardware connectivities.Concretely, we present preparation circuits for Dicke states $\left| {D_k^n} \right\rangle $ with (i) depth of O(klogkn) for all-to-all connectivity (such as on current ion trap devices); (ii) depth of O(k√kn)=O(nk) for Grid connectivity on grids of size Ω(√sn)×O(ns) with s ≤ k (such as on most current superconducting qubit devices).Both approaches have a total gate count of O(kn), need no ancilla qubits, and generalize to preparation and compression of those symmetric pure states, in which all non-zero amplitudes correspond to states with Hamming weight at most k. Our work significantly improves and expands previous state-of-the art circuits which had depth O(n) on a Linear Nearest Neighbor connectivity for arbitrary k (FCT 2019 [4]) and depth O(logn) on all-to-all connectivity for k = 1 (AQT 2019 [9]).