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arXiv:1401.8128 · Physical Review A

Implementing quantum control for unknown subroutines

Nicolai Friis, Vedran Dunjko, Wolfgang Dür, Hans J. Briegel

Published Mar 13, 2014• 99 citations
arXivHTMLPDF

Abstract

We present setups for the practical realization of adding control to unknown subroutines, supplementing the existing quantum optical scheme for black-box control with a counterpart for the quantum control of the ordering of sequences of operations. We also provide schemes to realize either task using trapped ions. These practical circumventions of recent no-go theorems are based on existing technologies. We argue that the possibility to add control to unknown operations in practice is a common feature of many physical systems. Based on the proposed implementations we discuss the apparent contradictions between theory and practice.

DOI
10.1103/physreva.89.030303

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We present setups for the practical realization of adding control to unknown subroutines, supplementing the existing quantum optical scheme for black-box control with a counterpart for the quantum control of the ordering of sequences of operations. We also provide schemes to realize either task using trapped ions. These practical circumventions of recent no-go theorems are based on existing technologies. We argue that the possibility to add control to unknown operations in practice is a common feature of many physical systems. Based on the proposed implementations we discuss the apparent contradictions between theory and practice.
One of the essential building blocks of classical computer programs is the ' if ' clause, which executes a subroutine depending on the value of a control variable. Similarly, several quantum algorithms rely on applying a unitary operation conditioned on the state of a control system. Here we show that this control cannot be performed by a quantum circuit if the unitary is completely unknown. The task remains impossible even if we allow the control to be done modulo a global phase. However, this no-go theorem does not prevent implementing quantum control of unknown unitaries in practice, as any physical implementation of an unknown unitary provides additional information that makes the control possible. We then argue that one should extend the quantum circuit formalism to capture this possibility in a straightforward way. This is done by allowing unknown unitaries to be applied to subspaces and not only to subsystems.
Modern programming relies on our ability to treat preprogrammed functions as black boxes - we can invoke them as subroutines without knowing their physical implementation. Here we show it is generally impossible to execute an unknown quantum subroutine. This, as a special case, forbids applying black-box subroutines conditioned on an ancillary qubit. We explore how this limits many quantum algorithms - forcing their circuit implementation to be individually tailored to specific inputs and inducing failure if these inputs are not known in advance. We present a method to avoid this situation for certain computational problems. We apply this method to enhance existing quantum factoring algorithms; reducing their complexity, and the extent to which they need to be tailored to factor specific numbers. Thus, we highlight a natural property of classical information that fails in the advent of quantum logic; and simultaneously demonstrate how to mitigate its effects in practical situations.
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.
Abstract In this work, we propose a generic protocol of distributed quantum computing when there are arbitrary unknown subroutines. It can be applied to scale up quantum computing by connecting a few quantum chips together, to the estimation task of unknown parameters or processes for the reduction of circuit depth, and also to quantum cryptography to construct secure protocols. Our protocol builds upon a few techniques we develop, such as the oblivious quantum teleportation and control, which can circumvent quantum no-go theorems on the manipulation of unknown objects. Furthermore, we demonstrate that this protocol can be physically implemented using currently available quantum computing platforms. These results suggest that our framework could provide a foundation for developing more advanced quantum algorithms and protocols in the future.
In this chapter, we shall discuss some practical applications of quantum control and dynamics. Our goal is not to give an exhaustive introduction to areas of application, but to point out the role of control and dynamics so as to further motivate the field as well as the techniques described in the previous chapters. The basics of mathematical modeling of quantum mechanical control systems were described in Chapter 2. Here we focus on more practical aspects in the areas of nuclear magnetic resonance and control of molecular and atomic systems. We also discuss a possible implementation of quantum information processing, namely the one with trapped ions. The dynamics here is used to implement quantum logic gates. We shall show how, changing the physical set-up, it is possible to create different Hamiltonians. The control is obtained by switching among these different Hamiltonians.
Realizing controlled operations is fundamental to the design and execution of quantum algorithms. In quantum simulation and learning of quantum many-body systems, an important subroutine consists of implementing a controlled Hamiltonian time-evolution. Given only black-box access to the uncontrolled evolution e−iHte^{-iHt}e−iHt, controlizing it, i.e., implementing ctrl(e−iHt)=∣0⟩⟨0∣⊗I+∣1⟩⟨1∣⊗e−iHt\mathrm{ctrl}(e^{-iHt}) = |0\rangle\langle 0|\otimes I + |1\rangle\langle 1 |\otimes e^{-iHt}ctrl(e−iHt)=∣0⟩⟨0∣⊗I+∣1⟩⟨1∣⊗e−iHt is non-trivial. Controlization has been recently used in quantum algorithms for transforming unknown Hamiltonian dynamics [OKTM24] leveraging a scheme introduced in Refs. [NSM15, DNSM21]. The main idea behind the scheme is to intersperse the uncontrolled evolution with suitable operations such that the overall dynamics approximates the desired controlled evolution. Although efficient, this scheme uses operations randomly sampled from an exponentially large set. In the present work, we show that more efficient controlization schemes can be constructed with the help of orthogonal arrays for unknown 2-local Hamiltonians. We conduct a detailed analysis of their performance and demonstrate the resulting improvements through numerical experiments. This construction can also be generalized to kkk-local Hamiltonians. Moreover, our controlization schemes based on orthogonal arrays can take advantage of the interaction graph's structure and be made more efficient.