[Paper Review] Quantum Algorithm for Generalized Deutsch-Jozsa Problem

Dong Pyo, Jinsoo Kim+1 more|ArXiv.org|May 16, 2000

Quantum Computing Algorithms and Architecture4 references7 citations

TL;DR

This paper generalizes the Deutsch-Jozsa problem to functions mapping Z_N to Z_M, introducing a quantum algorithm that determines whether a function is nonconstant or not evenly distributed using only a single evaluation. The algorithm employs a controlled phase kickback mechanism via a separable auxiliary register, avoiding the need for initialized ancilla qubits, and achieves correct classification by measuring the control register's output state after quantum Fourier transform and interference operations.

ABSTRACT

We generalize the Deutsch-Jozsa problem and present a quantum algorithm that can solve the generalized Deutsch-Jozsa problem by a single evaluation of a given function. We discuss the initialization of an auxiliary register and present a generalized Deutsch-Jozsa algorithm that requires no initialization of an auxiliary register.

Motivation & Objective

• To extend the original Deutsch-Jozsa problem to functions with outputs in Z_M rather than just Z_2.
• To develop a quantum algorithm that solves the generalized problem with only one function evaluation, maintaining quantum advantage.
• To eliminate the requirement for initializing the auxiliary register in the standard Deutsch-Jozsa setup.
• To generalize the concept of balanced functions to 'evenly distributed' functions over Z_M with equal-sized preimage sets.
• To demonstrate that the algorithm works even when the auxiliary register is prepared in a separable superposition state, not necessarily |−⟩.

Proposed method

• Generalizes the Deutsch-Jozsa problem by defining a function f: Z_N → Z_M as evenly distributed if it maps to K equally spaced values with equal preimage size ν = N/K.
• Uses a control register initialized in |0^n⟩ and an auxiliary register prepared in a general separable superposition |Ψ⟩ = ⊗_j (a_j|0⟩ + b_j|1⟩).
• Applies the sequence: W_n ⊗ I → U_f^⊕ → I ⊗ σ_z^⊗m → U_f^⊕ → I ⊗ σ_z^⊗m → W_n ⊗ I to induce a function-dependent phase shift.
• The phase shift is encoded via the parity function p: Z_M → Z_2^m, where f(x) is mapped to its binary representation and p(f(x)) computes the bitwise XOR of its bits.
• Employs quantum Fourier transform and interference to amplify differences between constant and evenly distributed functions in the final measurement.
• Shows that when f is constant, the final state collapses to |0^n⟩ with amplitude 1, while for evenly distributed f, the amplitude at |0^n⟩ is 0.

Experimental results

Research questions

• RQ1 Can the Deutsch-Jozsa algorithm be generalized to functions with non-binary outputs in Z_M?
• RQ2 Can the generalized problem be solved with only one function evaluation using quantum interference?
• RQ3 Is it possible to eliminate the need for initializing the auxiliary register in the generalized algorithm?
• RQ4 How should the concept of 'balanced' be extended to functions with outputs in Z_M?
• RQ5 What is the role of the phase kickback mechanism when the auxiliary register is not prepared in a Bell state?

Key findings

• The generalized algorithm determines whether f: Z_N → Z_M is nonconstant or not evenly distributed using only a single evaluation of f.
• When f is constant, the final state of the control register is |0^n⟩ with amplitude 1, and the measurement outcome is |0^n⟩ with certainty.
• When f is evenly distributed, the amplitude of |0^n⟩ in the final state is exactly 0, so the outcome |0^n⟩ never occurs.
• The algorithm remains valid even when the auxiliary register is prepared in a general separable superposition state, not requiring initialization to |−⟩.
• The method generalizes the phase kickback mechanism to arbitrary parity functions p: Z_M → Z_2^m, enabling function-dependent phase shifts without ancilla initialization.
• The algorithm achieves the same quantum advantage as the original Deutsch-Jozsa algorithm, solving the problem in one query while classical methods require up to N/2 + 1 evaluations in the worst case.