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Register a new userEfficient phase-factor evaluation in quantum signal processing
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Quantum signal processing (QSP) is a powerful quantum algorithm to exactly implement matrix polynomials on quantum computers. Asymptotic analysis of quantum algorithms based on QSP has shown that asymptotically optimal results can in principle be obtained for a range of tasks, such as Hamiltonian simulation and the quantum linear system problem. A further benefit of QSP is that it uses a minimal number of ancilla qubits, which facilitates its implementation on near-to-intermediate term quantum architectures. However, there is so far no classically stable algorithm allowing computation of the phase factors that are needed to build QSP circuits. Existing methods require the usage of variable precision arithmetic and can only be applied to polynomials of relatively low degree. We present here an optimization based method that can accurately compute the phase factors using standard double precision arithmetic operations. We demonstrate the performance of this approach with applications to Hamiltonian simulation, eigenvalue filtering, and the quantum linear system problems. Our numerical results show that the optimization algorithm can find phase factors to accurately approximate polynomials of degree larger than $10,000$ with error below $10^{-12}$.
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Quantum information offers the promise of being able to perform certain communication and computation tasks that cannot be done with conventional information technology (IT). Optical Quantum Information Processing (QIP) holds particular appeal, since it offers the prospect of communicating and computing with the same type of qubit. Linear optical techniques have been shown to be scalable, but the corresponding quantum computing circuits need many auxiliary resources. Here we present an alternative approach to optical QIP, based on the use of weak cross-Kerr nonlinearities and homodyne measurements. We show how this approach provides the fundamental building blocks for highly efficient non-absorbing single photon number resolving detectors, two qubit parity detectors, Bell state measurements and finally near deterministic control-not (CNOT) gates. These are essential QIP devices
Quantum logic gates with many control qubits are essential in many quantum algorithms, but remain challenging to perform in current experiments. Trapped ion quantum computers natively feature a different type of entangling operation, namely the Molmer-Sorensen (MS) gate which effectively applies an Ising interaction to all qubits at the same time. We consider a sequence of equal all-to-all MS operations, interleaved with single qubit gates that act only on one special qubit. Using a connection with quantum signal processing techniques, we find that it is possible to perform an arbitray SU(2) rotation on the special qubit if and only if all other qubits are in the state |1>. Such controlled rotation gates with N-1 control qubits require 2N applications of the MS gate, and can be mapped to a conventional Toffoli gate by demoting a single qubit to ancilla.
Enabled by rapidly developing quantum technologies, it is possible to network quantum systems at a much larger scale in the near future. To deal with non-Markovian dynamics that is prevalent in solid-state devices, we propose a general transfer function based framework for modeling linear quantum networks, in which signal flow graphs are applied to characterize the network topology by flow of quantum signals. We define a noncommutative ring $mathbb{D}$ and use its elements to construct Hamiltonians, transformations and transfer functions for both active and passive systems. The signal flow graph obtained for direct and indirect coherent quantum feedback systems clearly show the feedback loop via bidirectional signal flows. Importantly, the transfer function from input to output field is derived for non-Markovian quantum systems with colored inputs, from which the Markovian input-output relation can be easily obtained as a limiting case. Moreover, the transfer function possesses a symmetry structure that is analogous to the well-know scattering transformation in sd picture. Finally, we show that these transfer functions can be integrated to build complex feedback networks via interconnections, serial products and feedback, which may include either direct or indirect coherent feedback loops, and transfer functions between quantum signal nodes can be calculated by the Riegles matrix gain rule. The theory paves the way for modeling, analyzing and synthesizing non-Markovian linear quantum feedback networks in the frequency-domain.
We describe an algorithm for finding angle sequences in quantum signal processing, with a novel component we call halving based on a new algebraic uniqueness theorem, and another we call capitalization. We present both theoretical and experimental results that demonstrate the performance of the new algorithm. In particular, these two algorithmic ideas allow us to find sequences of more than 3000 angles within 5 minutes for important applications such as Hamiltonian simulation, all in standard double precision arithmetic. This is native to almost all hardware.
We show that many well-known signal transforms allow highly efficient realizations on a quantum computer. We explain some elementary quantum circuits and review the construction of the Quantum Fourier Transform. We derive quantum circuits for the Discrete Cosine and Sine Transforms, and for the Discrete Hartley transform. We show that at most O(log^2 N) elementary quantum gates are necessary to implement any of those transforms for input sequences of length N.
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