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arXiv:2210.07088 · Royal Society Open Science

‘Quantum supremacy’ revisited: low-complexity, deterministic solutions of the original Deutsch–Jozsa problem in classical physical systems

László B. Kish

Published Feb 28, 2023• 12 citations
arXivHTMLPDF

Abstract

The original Deutsch–Jozsa (oDJ) problem is for an oracle (realized here as a database) of size N , where, according to their claim, the deterministic solution of the problem on a classical Turing computer requires O( N ) computational complexity. They produced the famous Deutsch–Jozsa quantum algorithm that offered an exponential speed-up over the classical computer, namely O[log( N )] complexity for the solution in a quantum computer. In this paper, the problem is implemented on an instantaneous noise-based logic processor. It is shown that, similarly to the quantum algorithm, the oDJ problem can deterministically be solved with O[log( N )] complexity. The implication is that by adding a truly random coin to a classical Turing machine and using this classical-physical algorithm can also speed up the deterministic solution of the Deutsch–Jozsa problem exponentially, similarly to the quantum algorithm. Then it is realized that the same database and the solution of the Deutsch–Jozsa problem can also be realized by using an identical algorithmic structure in a simpler way, even without noise/random coin. The only lost function in this new system, as compared with noise-based logic, is the ability to do generic parallel logic operations over the whole database. As the latter feature is not needed for the oDJ problem, it is concluded that the problem can be solved on a classical computer with O[log( N )] complexity even without a random coin. Therefore, while the oDJ algorithm is a historically important step in the developments of quantum computers, it is insufficient to prove quantum supremacy. Note, there is also a simplified Deutsch–Jozsa problem proposed later, which is more popular in the field; however, it is irrelevant for the present paper.

DOI
10.1098/rsos.221327

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The original Deutsch–Jozsa (oDJ) problem is for an oracle (realized here as a database) of size N , where, according to their claim, the deterministic solution of the problem on a classical Turing computer requires O( N ) computational complexity. They produced the famous Deutsch–Jozsa quantum algorithm that offered an exponential speed-up over the classical computer, namely O[log( N )] complexity for the solution in a quantum computer. In this paper, the problem is implemented on an instantaneous noise-based logic processor. It is shown that, similarly to the quantum algorithm, the oDJ problem can deterministically be solved with O[log( N )] complexity. The implication is that by adding a truly random coin to a classical Turing machine and using this classical-physical algorithm can also speed up the deterministic solution of the Deutsch–Jozsa problem exponentially, similarly to the quantum algorithm. Then it is realized that the same database and the solution of the Deutsch–Jozsa problem can also be realized by using an identical algorithmic structure in a simpler way, even without noise/random coin. The only lost function in this new system, as compared with noise-based logic, is the ability to do generic parallel logic operations over the whole database. As the latter feature is not needed for the oDJ problem, it is concluded that the problem can be solved on a classical computer with O[log( N )] complexity even without a random coin. Therefore, while the oDJ algorithm is a historically important step in the developments of quantum computers, it is insufficient to prove quantum supremacy. Note, there is also a simplified Deutsch–Jozsa problem proposed later, which is more popular in the field; however, it is irrelevant for the present paper.
A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough. Two examples of quantum speed-up are the Deutsch–Jozsa and Simon’s problem, both efficiently solvable on a quantum Turing machine, and both believed to lack efficient classical solutions. Here we present a framework that can simulate both quantum algorithms efficiently, solving the Deutsch–Jozsa problem with probability 1 using only one oracle query, and Simon’s problem using linearly many oracle queries, just as expected of an ideal quantum computer. The presented simulation framework is in turn efficiently simulatable in a classical probabilistic Turing machine. This shows that the Deutsch–Jozsa and Simon’s problem do not require any genuinely quantum resources, and that the quantum algorithms show no speed-up when compared with their corresponding classical simulation. Finally, this gives insight into what properties are needed in the two algorithms and calls for further study of oracle separation between quantum and classical computation.
Instantaneous Noise-Based Logic (INBL) represents a computational paradigm that offers a deterministic alternative to quantum computing, potentially challenging the notion of quantum supremacy without relying on quantum hardware. INBL encodes logical information in orthogonal stochastic processes (“noise-bits”) and exploits their superpositions and nonlinear interactions to achieve an exponentially large computational space of dimension [Formula: see text], where M corresponds to the number of noise-bits analogous to qubits in quantum computing. This approach enables an exponential increase in computational throughput, with a computational speedup scaling on the order of O([Formula: see text], while maintaining hardware complexity comparable to quantum systems. Unlike quantum computers, INBL operates without decoherence, error correction, or probabilistic measurement, yielding deterministic outputs with low error probability. Demonstrated applications include exponential speed-gain compared to classical computers, such as INBL phonebook searches (for number or name lookup) and the implementation of the Deutsch–Jozsa algorithm, illustrating INBL’s capability to perform special-purpose computations with quantum-like exponential speedup using classical-physical noise-based hardware. We present an experimental comparison between the execution speeds of a Classical Turing machine algorithm — which changes the values of odd numbers in an exponentially large set to their next lower even numbers — and its INBL counterpart. Another experimental demonstration of the exponential speedup in finding and removing a given number from an exponentially large, unsorted set of integers.
A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough. Two examples of quantum speed-up are the Deutsch-Jozsa and Simon's problem, both efficiently solvable on a quantum Turing machine, and both believed to lack efficient classical solutions. Here we present a framework that can simulate both quantum algorithms efficiently, solving the Deutsch-Jozsa problem with probability 1 using only one oracle query, and Simon's problem using linearly many oracle queries, just as expected of an ideal quantum computer. The presented simulation framework is in turn efficiently simulatable in a classical probabilistic Turing machine. This shows that the Deutsch-Jozsa and Simon's problem do not require any genuinely quantum resources, and that the quantum algorithms show no speed-up when compared with their corresponding classical simulation. Finally, this gives insight into what properties are needed in the two algorithms, and calls for further study of oracle separation between quantum and classical computation.
Quantum Supremacy is an expression coined to indicate that there are some hard computational problems that quantum computers can solve exponentially faster than a classical-physical Turing machine (that is, a digital computer). Noise-based Logic (NBL) is a deterministic computation scheme that utilizes a digital computer and a physical true random number generator (TRNG). Recently, several computational problems have been identified where NBL also deterministically offered an exponential speedup compared to a digital computer. In the case of phonebook search, at proper conditions, NBL can even be exponentially faster than Grover's quantum algorithm that offers only a polynomial speedup compared to a digital computer. The newest result is the implementation of the Deutsch-Jozsa problem in NBL. The Deutsch-Jozsa quantum algorithm was historically the first one that was able to offer quantum supremacy. NBL offers the same speedup class. Open philosophical questions emerge about the role of true randomness in the speedup.
The Deutsch-Jozsa algorithm is one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm. It was proposed by Deutsch and Jozsa in 1992 with improvements by Cleve, Ekert, Macchiavello, and Mosca in 1998. The Deutsch-Jozsa problem is a promise problem and we can equivalently describe it as a partial function ${\text{DJ}}_{n}^{0}:{{0,1}}^{n}\ensuremath{\rightarrow}{0,1}$ defined as DJn0(x)=1{\text{DJ}}_{n}^{0}(x)=1DJn0​(x)=1 for ∣x∣=n/2,DJn0(x)=0|x|=n/2, {\text{DJ}}_{n}^{0}(x)=0∣x∣=n/2,DJn0​(x)=0 for ∣x∣=0,n|x|=0,n∣x∣=0,n, and it is undefined for the rest of the cases, where nnn is even, and ∣x∣|x|∣x∣ is the Hamming weight of xxx. The optimal quantum algorithm needs only one query to compute DJn0{\text{DJ}}_{n}^{0}DJn0​ but the classical deterministic algorithm requires ${2}^{n\ensuremath{-}1}+1$ queries to compute it in the worse case. In this article, we generalize the Deutsch-Jozsa problem as DJnk(x)=1{\text{DJ}}_{n}^{k}(x)=1DJnk​(x)=1 for ∣x∣=n/2,DJnk(x)=0|x|=n/2, {\text{DJ}}_{n}^{k}(x)=0∣x∣=n/2,DJnk​(x)=0 for ∣x∣|x|∣x∣ in the set ${0,1,...,k,n\ensuremath{-}k,n\ensuremath{-}k+1,...,n}$, and it is undefined for the rest of the cases, where $0\ensuremath{\le}k<n/2$. In particular, we give and prove an optimal exact quantum query algorithm with complexity k+1k+1k+1 for computing the generalized Deutsch-Jozsa problem DJnk{\text{DJ}}_{n}^{k}DJnk​. It is clear that the case of k=0k=0k=0 is in accordance with the Deutsch-Jozsa problem. Also, we give a method for finding the approximate and exact degrees of symmetric partial Boolean functions.
In 1985, David Deutsch challenged the Church-Turing thesis by stating that his quantum model of computation "could, in principle, be built and would have many remarkable properties not reproducible by any Turing machine". While this is thought to be true in general, there is usually no way of knowing that the corresponding classical algorithms are the best possible solutions. Here we provide an efficient classical simulation of the Deutsch-Jozsa algorithm, which was one of the first examples of quantum computational speed-up. Our conclusion is that the Deutsch-Jozsa quantum algorithm owes its speed-up to resources that are not necessarily quantum-mechanical, and when compared with the classical simulation offers no speed-up at all.