#quantumwavedivider

DQC: Dual Quantum Computing Duality DQC distinguishes itself from traditional quantum computers with particle-wave duality. Unlike normal quantum computers, DQC permits linear unitary operations. Non-unitary generalised quantum gates or duality gates can operate. This unique method makes quantum algorithm construction more versatile and may speed up computer processes.

History GUI-Lu Long introduced Duality Quantum Computing in 2002. It appeared in an October 2002 SPIE abstract. Long's idea employed quantum interference to compute with quantum systems' wave nature.

Works How

Quantum wave functions are separated and combined by DQC. This is like a quantum computer “on the move and passing through a multi-slit”. A typical DQC computation involves four steps: A wave function representing the initial quantum state is broken into multiple “sub-waves” or attenuated, identical sections by QWD. Like a quantum system in a multi-slit, its wave function splits into d sub-waves with the same internal wave function but differing centre of mass movements. Division does not contradict the no-cloning theorem because it divides the same quantum system's wave function. The divider operator Dp is a linear isometry that creates attenuated copies of a state ψ. Quantum gates (unitary operations) vary per sub-wave. Beyond quantum parallelism, duality parallelism allows sub-wave function gate operations at different slits. This allows linear combination of unitaries. After passing through quantum gates, sub-waves form a single wave function. Interference from recombination encodes computation outcomes. Combiner Cp reverses divider effect. Finally measuring the composite wave function produces the computational output. A “slit” (e.g., the 0-slit) is used in most single-output duality quantum computing measurements. Architecture and Features

Conceptual DQC architecture incorporates a QWD and wave combiner. This is necessary for operation. The main DQC characteristics are:

Division, parallel unitary processes, and combination produce a duality gate or generalised quantum gate. The gates are frequently non-unitary. Generalest duality gate For unitary operators Ui with complex coefficients ci, Lc = Σ_{i=0}^{d-1} ci Ui, where |ci| ≤ 1 and |Σ ci| ≤ 1. In finite-dimensional Hilbert space, generalised quantum gates can realise linear bounded operators. Generalised quantum gates have unitary operators as “extreme points”. Building a “moving quantum computer passing through a multi-slit” DQC is experimentally tough. To imitate a duality computer, a normal quantum computer with an extra qubit resource was created. An n-qubit quantum computer with a qudit can simulate a duality computer in d-slits. Some believe a duality computer and a quantum computer in duality mode have equivalent processing power. Due to its low qubit count, duality mode is easy to implement experimentally. Both modalities convey divider operation, but duality has a deeper, normalised combiner. The mathematical theory of DQC is well-studied. Dividers, combiners, and generalised quantum gates are important. WDD limits what cannot be a generalised quantum gate to define duality computing workloads. A generalised quantum gate cannot perturb a semi-Fredholm partial isometry with a non-zero Fredholm index to a finite rank in infinite dimension. The non-unitary operations of DQC make it suitable for implementing Kraus operators, which explain open quantum system dynamics. Applications

Applications that use linear operator combinations benefit from DQC: Database Search: DQC may identify a marked item in an unsorted database faster than Grover's technique with a single query. While classical search may take O(N) steps, it uses substantially fewer qubit resources (log₂N) than classical computation (N log₂N). Initial theoretical studies suggest DQC can handle NP-complete problems with polynomials. Later conclusions demonstrated that these cases contradict mainstream quantum physics and should be rejected. Yet duality beats quantum computers. Quantum system ground state preparation using DQC may conserve qubit resources. DQC is useful for simulating non-unitary evolutions in open quantum systems because it can actualise Kraus operators through linear combinations of unitary operators. Every iteration is quadratic faster than classical algorithms and exponentially more exact than unitary simulation. Compared to classical methods, DQC algorithms accelerate PDEs quadratically per iteration. DQC simulates quantum systems with sparse Hamiltonians with exponential precision using Childs and Wiebe and Berry et al. techniques. Classical factorisation can be converted into duality computing algorithms to save qubits. DQC affects quantum operations' convexity and mathematical theory. It can predict the translation of most classical algorithms into quantum algorithms via duality mode, allowing both to run on the same quantum computer. Types

Superconducting and trapped-ion quantum computers are different from DQC. Instead, it is a computational model or theoretical framework for several physical systems. The “giant molecule scheme” and “nonlinear quantum optics scheme” are proposed physical architectures. A traditional quantum computer with additional qubits in duality mode is best for DQC simulation. Advantages

Enhanced Computing Operations: DQC has more quantum division and combiner operations than classical quantum computers. It inherently favours non-unitary evolution in open quantum systems. DQC is a revolutionary technique to construct quantum algorithms that can work with classical and quantum algorithms. DQC may be faster for linearly coupled unitary jobs than traditional and ordinary quantum computers. Open quantum systems are simulated with quadratic acceleration and exponential precision. Resource Efficiency: DQC algorithms, including unsorted database search, use fewer qubits than traditional computation (e.g., log₂N vs. N log₂N). Cons and Issues