Efficient networks for quantum factoring

Artificial intelligence, human cognition, and conscious supremacy

The computational significance of consciousness is an important and potentially more tractable research theme than the hard problem of consciousness, as one could look at the correlation of consciousness and computational capacities through, e.g., algorithmic or complexity analyses. In the literature, consciousness is defined as what it is like to be an agent (i.e., a human or a bat), with phenomenal properties, such as qualia, intentionality, and self-awareness. The absence of these properties would be termed "unconscious." The recent success of large language models (LLMs), such as ChatGPT, has raised new questions about the computational significance of human conscious processing. Although instances from biological systems would typically suggest a robust correlation between intelligence and consciousness, certain states of consciousness seem to exist without manifest existence of intelligence. On the other hand, AI systems seem to exhibit intelligence without consciousness. These instances seem to suggest possible dissociations between consciousness and intelligence in natural and artificial systems. Here, I review some salient ideas about the computational significance of human conscious processes and identify several cognitive domains potentially unique to consciousness, such as flexible attention modulation, robust handling of new contexts, choice and decision making, cognition reflecting a wide spectrum of sensory information in an integrated manner, and finally embodied cognition, which might involve unconscious processes as well. Compared to such cognitive tasks, characterized by flexible and ad hoc judgments and choices, adequately acquired knowledge and skills are typically processed unconsciously in humans, consistent with the view that computation exhibited by LLMs, which are pretrained on a large dataset, could in principle be processed without consciousness, although conversations in humans are typically done consciously, with awareness of auditory qualia as well as the semantics of what are being said. I discuss the theoretically and practically important issue of separating computations, which need to be conducted consciously from those which could be done unconsciously, in areas, such as perception, language, and driving. I propose conscious supremacy as a concept analogous to quantum supremacy, which would help identify computations possibly unique to consciousness in biologically practical time and resource limits. I explore possible mechanisms supporting the hypothetical conscious supremacy. Finally, I discuss the relevance of issues covered here for AI alignment, where computations of AI and humans need to be aligned.

Computing Shor's algorithmic steps with interference patterns of classical light

When considered as orthogonal bases in distinct vector spaces, the unit vectors of polarization directions and the Laguerre-Gaussian modes of polarization amplitude are inseparable, constituting a so-called classical entangled light beam. Equating this classical entanglement to quantum entanglement necessary for computing purpose, we show that the parallelism featured in Shor's factoring algorithm is equivalent to the concurrent light-path propagation of an entangled beam or pulse train. A gedanken experiment is proposed for executing the key algorithmic steps of modular exponentiation and Fourier transform on a target integer N using only classical manipulations on the amplitudes and polarization directions. The multiplicative order associated with the sought-after integer factors is identified through a four-hole diffraction interference from sources obtained from the entangled beam profile. The unique mapping from the fringe patterns to the computed order is demonstrated through simulations for the case [Formula: see text].

Coherence as a Resource for Shor's Algorithm

Shor's factoring algorithm provides a superpolynomial speedup over all known classical factoring algorithms. Here, we address the question of which quantum properties fuel this advantage. We investigate a sequential variant of Shor's algorithm with a fixed overall structure and identify the role of coherence for this algorithm quantitatively. We analyze this protocol in the framework of dynamical resource theories, which capture the resource character of operations that can create and detect coherence. This allows us to derive a lower and an upper bound on the success probability of the protocol, which depend on rigorously defined measures of coherence as a dynamical resource. We compare these bounds with the classical limit of the protocol and conclude that within the fixed structure that we consider, coherence is the quantum resource that determines its performance by bounding the success probability from below and above. Therefore, we shine new light on the fundamental role of coherence in quantum computation.

Demonstration of Shor's factoring algorithm for N [Formula: see text] 21 on IBM quantum processors

We report a proof-of-concept demonstration of a quantum order-finding algorithm for factoring the integer 21. Our demonstration involves the use of a compiled version of the quantum phase estimation routine, and builds upon a previous demonstration. We go beyond this work by using a configuration of approximate Toffoli gates with residual phase shifts, which preserves the functional correctness and allows us to achieve a complete factoring of [Formula: see text]. We implemented the algorithm on IBM quantum processors using only five qubits and successfully verified the presence of entanglement between the control and work register qubits, which is a necessary condition for the algorithm's speedup in general. The techniques we employ may be useful in carrying out Shor's algorithm for larger integers, or other algorithms in systems with a limited number of noisy qubits.

Proof-of-principle demonstration of compiled Shor's algorithm using a quantum dot single-photon source

We report a proof-of-principle demonstration of Shor's algorithm with photons generated by an on-demand semiconductor quantum dot single-photon source for the first time. A fully compiled version of Shor's algorithm for factoring 15 has been accomplished with a significantly reduced resource requirement that employs the four-photon cluster state. Genuine multiparticle entanglement properties are confirmed to reveal the quantum character of the algorithm and circuit. The implementation realizes the Shor's algorithm with deterministic photonic qubits, which opens new applications for cluster state beyond one-way quantum computing.

Quantum Simulation Logic, Oracles, and the Quantum Advantage

Query complexity is a common tool for comparing quantum and classical computation, and it has produced many examples of how quantum algorithms differ from classical ones. Here we investigate in detail the role that oracles play for the advantage of quantum algorithms. We do so by using a simulation framework, Quantum Simulation Logic (QSL), to construct oracles and algorithms that solve some problems with the same success probability and number of queries as the quantum algorithms. The framework can be simulated using only classical resources at a constant overhead as compared to the quantum resources used in quantum computation. Our results clarify the assumptions made and the conditions needed when using quantum oracles. Using the same assumptions on oracles within the simulation framework we show that for some specific algorithms, such as the Deutsch-Jozsa and Simon’s algorithms, there simply is no advantage in terms of query complexity. This does not detract from the fact that quantum query complexity provides examples of how a quantum computer can be expected to behave, which in turn has proved useful for finding new quantum algorithms outside of the oracle paradigm, where the most prominent example is Shor’s algorithm for integer factorization.

Feasibility of encoding Shor's algorithm into the motional states of an ion in the anharmonic trap

We demonstrate theoretically that it may be possible to encode states of a multi-qubit system into the progression of quantized motional∕vibrational levels of an ion trapped in a weakly anharmonic potential. Control over such register of quantum information is achieved by applying oscillatory radio-frequency fields shaped optimally for excitation of the desired state-to-state transitions. Anharmonicity of the vibrational spectrum plays a key role in this approach to the control and quantum computation, since it allows resolving different state-to-state transitions and addressing them selectively. Optimal control theory is used to derive pulses for implementing the four-qubit version of Shor's algorithm in a single step. Accuracy of the qubit-state transformations, reached in the numerical simulations, is around 0.999. Very detailed insight is obtained by analysis of the time-evolution of state populations and by spectral analysis of the optimized pulse.

Adding control to arbitrary unknown quantum operations

Although quantum computers promise significant advantages, the complexity of quantum algorithms remains a major technological obstacle. We have developed and demonstrated an architecture-independent technique that simplifies adding control qubits to arbitrary quantum operations—a requirement in many quantum algorithms, simulations and metrology. The technique, which is independent of how the operation is done, does not require knowledge of what the operation is, and largely separates the problems of how to implement a quantum operation in the laboratory and how to add a control. Here, we demonstrate an entanglement-based version in a photonic system, realizing a range of different two-qubit gates with high fidelity.

Quantum computing has advantages over conventional computing, but the complexity of quantum algorithms creates technological challenges. Here, an architecture-independent technique, that simplifies adding control qubits to arbitrary quantum operations, is developed and demonstrated.

Demonstration of a compiled version of Shor's quantum factoring algorithm using photonic qubits

We report an experimental demonstration of a complied version of Shor's algorithm using four photonic qubits. We choose the simplest instance of this algorithm, that is, factorization of N=15 in the case that the period r=2 and exploit a simplified linear optical network to coherently implement the quantum circuits of the modular exponential execution and semiclassical quantum Fourier transformation. During this computation, genuine multiparticle entanglement is observed which well supports its quantum nature. This experiment represents an essential step toward full realization of Shor's algorithm and scalable linear optics quantum computation.

Experimental demonstration of a compiled version of Shor's algorithm with quantum entanglement

Shor's powerful quantum algorithm for factoring represents a major challenge in quantum computation. Here, we implement a compiled version in a photonic system. For the first time, we demonstrate the core processes, coherent control, and resultant entangled states required in a full-scale implementation. These are necessary steps on the path towards scalable quantum computing. Our results highlight that the algorithm performance is not the same as that of the underlying quantum circuit and stress the importance of developing techniques for characterizing quantum algorithms.

Model based on a quantum algorithm to study the evolution of an epidemics

A model based on a quantum algorithm is used to study the spread of HIV virus and to predict infection rates on individuals who are not aware of their particular condition. The model makes an analogy between quantum systems and individuals who are infected by the disease. Individuals are represented by two-level quantum systems (quantum "bit"), and the interactions among individuals who cause the infection are represented by unitary transformations. The population is divided into categories according to their behaviour, and the interactions among those individuals in the same category and those in different categories are simulated. The objective is to obtain statistical data on the number of infected individuals depending on the time for every category and for the entire population. Besides, we analyse the impact of the evolution of the disease on individuals who have not knowledge of their specific sanitary condition.

Quantum networks

As classical information technology approaches limits of size and functionality, practitioners are searching for new paradigms for the distribution and processing of information. Our goal in this Introduction is to provide a broad view of the beginning of a new era in information technology, an era of quantum information, where previously underutilized quantum effects, such as quantum superposition and entanglement, are employed as resources for information encoding and processing. The ability to distribute these new resources and connect distant quantum systems will be critical. We present an overview of network implications for quantum communication applications, and for quantum computing. This overview is a selection of several illustrative examples, to serve as motivation for the network research community to bring its expertise to the development of quantum information technologies.

Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance

The number of steps any classical computer requires in order to find the prime factors of an l-digit integer N increases exponentially with l, at least using algorithms known at present. Factoring large integers is therefore conjectured to be intractable classically, an observation underlying the security of widely used cryptographic codes. Quantum computers, however, could factor integers in only polynomial time, using Shor's quantum factoring algorithm. Although important for the study of quantum computers, experimental demonstration of this algorithm has proved elusive. Here we report an implementation of the simplest instance of Shor's algorithm: factorization of N = 15 (whose prime factors are 3 and 5). We use seven spin-1/2 nuclei in a molecule as quantum bits, which can be manipulated with room temperature liquid-state nuclear magnetic resonance techniques. This method of using nuclei to store quantum information is in principle scalable to systems containing many quantum bits, but such scalability is not implied by the present work. The significance of our work lies in the demonstration of experimental and theoretical techniques for precise control and modelling of complex quantum computers. In particular, we present a simple, parameter-free but predictive model of decoherence effects in our system.

Stable quantum computation of unstable classical chaos

We show on the example of the Arnold cat map that classical chaotic systems can be simulated with exponential efficiency on a quantum computer. Although classical computer errors grow exponentially with time, the quantum algorithm with moderate imperfections is able to simulate accurately the unstable chaotic classical nonlinear dynamics for long times. The algorithm can be easily implemented on systems of a few qubits.

Exponential gain in quantum computing of quantum chaos and localization

We present a quantum algorithm which simulates the quantum kicked rotator model exponentially faster than classical algorithms. This shows that important physical problems of quantum chaos, localization, and Anderson transition can be modeled efficiently on a quantum computer. We also show that a similar algorithm simulates efficiently classical chaos in certain area-preserving maps.

A scalable quantum computer with ions in an array of microtraps

Quantum computers require the storage of quantum information in a set of two-level systems (called qubits), the processing of this information using quantum gates and a means of final readout. So far, only a few systems have been identified as potentially viable quantum computer models--accurate quantum control of the coherent evolution is required in order to realize gate operations, while at the same time decoherence must be avoided. Examples include quantum optical systems (such as those utilizing trapped ions or neutral atoms, cavity quantum electrodynamics and nuclear magnetic resonance) and solid state systems (using nuclear spins, quantum dots and Josephson junctions). The most advanced candidates are the quantum optical and nuclear magnetic resonance systems, and we expect that they will allow quantum computing with about ten qubits within the next few years. This is still far from the numbers required for useful applications: for example, the factorization of a 200-digit number requires about 3,500 qubits, rising to 100,000 if error correction is implemented. Scalability of proposed quantum computer architectures to many qubits is thus of central importance. Here we propose a model for an ion trap quantum computer that combines scalability (a feature usually associated with solid state proposals) with the advantages of quantum optical systems (in particular, quantum control and long decoherence times).