Ancillary Qubits Research Articles

Spectroscopic estimation of the photon number for superconducting Kerr parametric oscillators

Quantum annealing (QA) is a way to solve combinational optimization problems. Kerr nonlinear parametric oscillators (KPOs) are promising devices for implementing QA. When we solve the combinational optimization problems using KPOs, it is necessary to precisely control the photon number of the KPOs. Here, we propose a feasible method to estimate the photon number of the KPO. We consider coupling an ancillary qubit to the KPO and show that spectroscopic measurements on the ancillary qubit provide information on the photon number of the KPO.

Analytical Formulation of the Second-Order Derivative of Energy for the Orbital-Optimized Variational Quantum Eigensolver: Application to Polarizability.

We develop a quantum-classical hybrid algorithm to calculate the analytical second-order derivative of the energy for the orbital-optimized variational quantum eigensolver (OO-VQE), which is a method to calculate eigenenergies of a given molecular Hamiltonian by utilizing near-term quantum computers and classical computers. We show that all quantities required in the algorithm to calculate the derivative can be evaluated on quantum computers as standard quantum expectation values without using any ancillary qubits. We validate our formula by numerical simulations of quantum circuits for computing the polarizability of the water molecule, which is the second-order derivative of the energy, with respect to the electric field. Moreover, the polarizabilities and refractive indices of thiophene and furan molecules are calculated as a test bed for possible industrial applications. We finally analyze the error scaling of the estimated polarizabilities obtained by the proposed analytical derivative versus the numerical derivative obtained by the finite difference. Numerical calculations suggest that our analytical derivative requires fewer measurements (runs) on quantum computers than the numerical derivative to achieve the same fixed accuracy.

Optimal (controlled) quantum state preparation and improved unitary synthesis by quantum circuits with any number of ancillary qubits

As a cornerstone for many quantum linear algebraic and quantum machine learning algorithms, controlled quantum state preparation (CQSP) aims to provide the transformation of |i⟩|0n⟩→|i⟩|ψi⟩ for all i∈{0,1}k for the given n-qubit states |ψi⟩. In this paper, we construct a quantum circuit for implementing CQSP, with depth O(n+k+2n+kn+k+m) and size O(2n+k) for any given number m of ancillary qubits. These bounds, which can also be viewed as a time-space tradeoff for the transformation, are optimal for any integer parameters m,k≥0 and n≥1. When k=0, the problem becomes the canonical quantum state preparation (QSP) problem with ancillary qubits, which asks for efficient implementations of the transformation |0n⟩|0m⟩→|ψ⟩|0m⟩. This problem has many applications with many investigations, yet its circuit complexity remains open. Our construction completely solves this problem, pinning down its depth complexity to Θ(n+2n/(n+m)) and its size complexity to Θ(2n) for any m. Another fundamental problem, unitary synthesis, asks to implement a general n-qubit unitary by a quantum circuit. Previous work shows a lower bound of Ω(n+4n/(n+m)) and an upper bound of O(n2n) for m=Ω(2n/n) ancillary qubits. In this paper, we quadratically shrink this gap by presenting a quantum circuit of the depth of O(n2n/2+n1/223n/2m1/2  ).

Restoring broken symmetries using quantum search “oracles”

We present a new method to perform variation after projection in many-body systems on quantum computers that does not require performing explicit projection. The technique employs the notion of ``oracle'', generally used in quantum search algorithms. We show how to construct the oracle and the projector associated with a symmetry operator. The procedure is illustrated for the parity, particle number, and total spin symmetries. The oracle is used to restore symmetry by indirect measurements using a single ancillary qubit. An illustration of the technique is made to obtain the approximate ground state energy for the pairing model Hamiltonian.

Iteration-free digital quantum simulation of imaginary-time evolution based on the approximate unitary expansion

Imaginary-time evolution plays an important role in many areas of quantum physics and has been widely applied to the ground-state determination of various Hamiltonian in the quantum computation field. In this work, we propose an iteration-free quantum algorithm in a full gate-based frame using the approximate unitary expansion to simulate the imaginary-time evolution operator, avoiding the resource overhead caused by repeated measurement for state reconstruction or complex pre-calculations in the classical computers. We detail the algorithm and analyze the complexity and related characteristics including a lower bound for ancillary qubits at a given success probability. Then an application demonstration of the algorithm in quantum chemistry with hydrogen molecule under noiseless and noisy conditions is offered. In addition, we present another imaginary-time evolution simulation method based on similar construction schemes. Our algorithms can serve as the alternative proposals for the imaginary-time evolution realization in the future fault-tolerant quantum computers.

Configurable sublinear circuits for quantum state preparation

The theory of quantum algorithms promises unprecedented benefits of harnessing the laws of quantum mechanics for solving certain computational problems. A persistent obstacle to using such algorithms for solving a wide range of real-world problems is the cost of loading classical data to a quantum state. Several quantum circuit-based methods have been proposed for encoding classical data as probability amplitudes of a quantum state. However, they require either quantum circuit depth or width to grow linearly with the data size, even though the other dimension of the quantum circuit grows logarithmically. In this paper, we present a configurable bidirectional procedure that addresses this problem by tailoring the resource trade-off between quantum circuit width and depth. In particular, we show a configuration that encodes an $N$-dimensional state by a quantum circuit with $O(\sqrt{N})$ width and depth and entangled information in ancillary qubits. We show a proof-of-principle on five quantum computers and compare the results.

Generalized Toffoli Gate Decomposition Using Ququints: Towards Realizing Grover's Algorithm with Qudits.

Qubits, which are the quantum counterparts of classical bits, are used as basic information units for quantum information processing, whereas underlying physical information carriers, e.g., (artificial) atoms or ions, admit encoding of more complex multilevel states-qudits. Recently, significant attention has been paid to the idea of using qudit encoding as a way for further scaling quantum processors. In this work, we present an efficient decomposition of the generalized Toffoli gate on five-level quantum systems-so-called ququints-that use ququints' space as the space of two qubits with a joint ancillary state. The basic two-qubit operation we use is a version of the controlled-phase gate. The proposed N-qubit Toffoli gate decomposition has O(N) asymptotic depth and does not use ancillary qubits. We then apply our results for Grover's algorithm, where we indicate on the sizable advantage of using the qudit-based approach with the proposed decomposition in comparison to the standard qubit case. We expect that our results are applicable for quantum processors based on various physical platforms, such as trapped ions, neutral atoms, protonic systems, superconducting circuits, and others.

Optimization of CNOT circuits on limited-connectivity architecture

A CNOT circuit is the key gadget for entangling qubits in quantum computing systems. However, the qubit connectivity of noisy intermediate-scale quantum (NISQ) devices is constrained by their {limited connectivity architecture}. To improve the performance of CNOT circuits on NISQ devices, we investigate the optimization of the size/depth of CNOT circuits under the limited connectivity architecture. We present a method that can optimize the size of any $n$-qubit CNOT circuit $O\left(\frac{n^2}{\log \delta}\right)$ on any connected graph with minimum degree $\delta$, and prove this bound is optimal for the regular graph. For the near-term sparsely connected structure, we additionally present a method that can optimize the size of any $n$-qubit CNOT circuit to below $2n^2$. The numerical experiment shows that our method performs better than state-of-the-art results. Specifically, we present an example to illustrate the applicability of our algorithm. For the grid structure, which is commonly used in current quantum devices, we demonstrate that the depth of any $n$-qubit CNOT circuit can be optimized to be linear in $n$ with certain ancillary qubits (ancillas). Experimental results indicate that this method has significant improvements compared with all of the existing methods. We additionally test our algorithms on the five-qubit IBMQ devices, and the experiments show that the measurement results of the optimized circuit with our algorithm are more robust to noise compared with the IBM mapping method.

Breakthrough of error correction in quantum computing

By using a superconducting processor, Google Quantum AI group demonstrated that a logic qubit realized by surface code of quantum error correction performs better when the number of physical qubits increases. Two surface codes to encode logic qubits for scaling are realized experimentally, with multiple cycles of quantum error correcting assisted by ancillary qubits. This result can be considered as an important step toward fault-tolerant quantum computers. In this review paper, we introduce briefly the mechanism of quantum error correction. As an example, Shor’s nine-qubit error correction code is explained. Then, the new experiments of Google quantum AI group are introduced to show their significance in scaling. The advances in other quantum error correction experiments are also reviewed. Finally, the development of quantum computers is discussed.

Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications.

Quantum state preparation is an important subroutine for quantum computing. We show that any n-qubit quantum state can be prepared with a Θ(n)-depth circuit using only single- and two-qubit gates, although with a cost of an exponential amount of ancillary qubits. On the other hand, for sparse quantum states with d⩾2 nonzero entries, we can reduce the circuit depth to Θ(log(nd)) with O(ndlogd) ancillary qubits. The algorithm for sparse states is exponentially faster than best-known results and the number of ancillary qubits is nearly optimal and only increases polynomially with the system size. We discuss applications of the results in different quantum computing tasks, such as Hamiltonian simulation, solving linear systems of equations, and realizing quantum random access memories, and find cases with exponential reductions of the circuit depth for all these three tasks. In particular, using our algorithm, we find a family of linear system solving problems enjoying exponential speedups, even compared to the best-known quantum and classical dequantization algorithms.

Rapid solution of logical equivalence problems by quantum computation algorithm

We present a quantum computation algorithm that enables solving the problem of logical equivalence verification in exponentially less time than the classical deterministic computation. In this novel quantum algorithm, the oracles of the two evaluated functions are executed in series to yield a common target qubit which then interacts with an ancillary qubit. We found that the degree of entanglement (measured by the concurrence) of the target and ancillary qubits is a reliable witness for the logical equivalence property of the two functions. The steps number of the quantum algorithm is inversely proportional to the square of the standard error ϵ2 of the measured concurrence value, with no dependence on the input size ′n′ of each function. This corresponds to a number of evaluations of the two functions: O(ϵ−2) for the quantum algorithm compared with O(2n) for the classical approach. To assess the algorithm performance, two sets of experiments are conducted using the IBM Q Experience simulator for input sizes: 2 and 12 variables per function. While the former verifies that the results of the experiment are in a good match with the theory, the latter showcases the quantum supremacy of the presented algorithm. In particular, The latter shows that the quantum algorithm requires only 200 oracles queries compared with 213 queries for the classical algorithm.

Randomized Ancillary Qubit Overcomes Detector-Control and Intercept-Resend Hacking of Quantum Key Distribution

Practical implementations of quantum key distribution (QKD) have been shown to be subject to various detector side-channel attacks that compromise the promised unconditional security. Most notable is a general class of attacks adopting the use of faked-state photons as in the detector-control and, more broadly, the intercept-resend attacks. In this paper, we present a simple scheme to overcome such class of attacks: A legitimate user, Bob, uses a polarization randomizer at his gateway to distort an ancillary polarization of a phase-encoded photon in a bidirectional QKD configuration. Passing through the randomizer once on the way to his partner, Alice, and again in the opposite direction, the polarization qubit of the genuine photon is immune to randomization. However, the polarization state of a photon from an intruder, Eve, to Bob is randomized and hence directed to a detector in a different path, whereupon it triggers an alert. We demonstrate theoretically and experimentally that, using commercial off-the-shelf detectors, it can be made impossible for Eve to avoid triggering the alert, no matter what faked-state of light she uses.

Quantum phase measurement of two-qubit states in an open waveguide

We present a new method for quantum state tomography within a single-excitation subspace of two-qubit states in an open waveguide. The system under investigation consists of three qubits in an open waveguide, separated by a distance comparable to the wavelength of the electromagnetic field. We show that the modulation of the frequency of the central ancillary qubit allows us to obtain unambiguous information about the initial phase difference $\varphi_1-\varphi_3$ of the edge qubits via the measurement of the evolution of their probability amplitudes.

SWAP test for an arbitrary number of quantum states

We develop a recursive algorithm to generalize the quantum SWAP test for an arbitrary number m of quantum states requiring O(m) controlled-swap (CSWAP) gates and $$O(\log m)$$ ancillary qubits. We construct a quantum circuit able to simultaneously measure overlaps $$|\langle \phi _i, \phi _j\rangle |^2$$ of m arbitrary pure states $${|{\phi _1\ldots \phi _m}\rangle }$$ . Our construction relies on a pairing unitary that generates a superposition state where every pair of input states is labeled by a basis state formed by the ancillaries. By implementing a simple genetic algorithm, we give numerical evidence indicating that our method of labeling each pair of inputs using CSWAP gates is optimal up to $$m=8$$ . Potential applications of the new circuits in the context of quantum machine learning are discussed.

Linear-depth quantum circuits for multiqubit controlled gates

Quantum circuit depth minimization is critical for practical applications of circuit-based quantum computation. In this work, we present a systematic procedure to decompose multiqubit controlled unitary gates, which is essential in many quantum algorithms, to controlled-NOT and single-qubit gates with which the quantum circuit depth only increases linearly with the number of control qubits. Our algorithm does not require any ancillary qubits and achieves a quadratic reduction of the circuit depth against known methods. We show the advantage of our algorithm with proof-of-principle experiments on the IBM quantum cloud platform.

Variational Entanglement-Assisted Quantum Process Tomography with Arbitrary Ancillary Qubits.

Quantum process tomography is a pivotal technique in fully characterizing quantum dynamics. However, exponential scaling of the Hilbert space with the increasing system size extremely restrains its experimental implementations. Here, we put forward a more efficient, flexible, and error-mitigated method: variational entanglement-assisted quantum process tomography with arbitrary ancillary qubits. Numerically, we simulate up to eight-qubit quantum processes and show that this tomography with m ancillary qubits (0≤m≤n) alleviates the exponential costs on state preparation (from 4^{n} to 2^{n-m}), measurement settings (at least a 1 order of magnitude reduction), and data postprocessing (efficient and robust parameter optimization). Experimentally, we first demonstrate our method on a silicon photonic chip by rebuilding randomly generated one-qubit and two-qubit unitary quantum processes. Further using the error mitigation method, two-qubit quantum processes can be rebuilt with average gate fidelity enhanced from 92.38% to 95.56%. Our Letter provides an efficient and practical approach to process tomography on the noisy quantum computing platforms.

Quantum simulation of nonequilibrium dynamics and thermalization in the Schwinger model

We present simulations of nonequilibrium dynamics of quantum field theories on digital quantum computers. As a representative example, we consider the Schwinger model, a ($1+1$)-dimensional U(1) gauge theory, coupled through a Yukawa-type interaction to a thermal environment described by a scalar field theory. We use the Hamiltonian formulation of the Schwinger model discretized on a spatial lattice. With the thermal scalar fields traced out, the Schwinger model can be treated as an open quantum system and its real-time dynamics are governed by a Lindblad equation in the Markovian limit. The interaction with the environment ultimately drives the system to thermal equilibrium. In the quantum Brownian motion limit, the Lindblad equation is related to a field theoretical Caldeira-Leggett equation. By using the Stinespring dilation theorem with ancillary qubits, we perform studies of both the nonequilibrium dynamics and the preparation of a thermal state in the Schwinger model using IBM's simulator and quantum devices. The real-time dynamics of field theories as open quantum systems and the thermal state preparation studied here are relevant for a variety of applications in nuclear and particle physics, quantum information and cosmology.

Equivalence Checking of Quantum Circuits With the ZX-Calculus

As state-of-the-art quantum computers are capable of running increasingly complex algorithms, the need for automated methods to design and test potential applications rises. Equivalence checking of quantum circuits is an important, yet hardly automated, task in the development of the quantum software stack. Recently, new methods have been proposed that tackle this problem from widely different perspectives. One of them is based on the ZX-calculus, a graphical rewriting system for quantum computing. However, the power and capability of this equivalence checking method has barely been explored. The aim of this work is to evaluate the ZX-calculus as a tool for equivalence checking of quantum circuits. To this end, it is demonstrated how the ZX-calculus based approach for equivalence checking can be expanded in order to verify the results of compilation flows and optimizations on quantum circuits. It is also shown that the ZX-calculus based method is not complete—especially for quantum circuits with ancillary qubits. In order to properly evaluate the proposed method, we conduct a detailed case study by comparing it to two other state-of-the-art methods for equivalence checking: one based on path-sums and another based on decision diagrams. The proposed methods have been integrated into the publicly available QCEC tool ( <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><uri>https://github.com/cda-tum/qcec</uri></monospace> ) which is part of the Munich Quantum Toolkit (MQT).

Experimental Realization of a Quantum Refrigerator Driven by Indefinite Causal Orders.

Indefinite causal order (ICO) is playing a key role in recent quantum technologies. Here, we experimentally study quantum thermodynamics driven by ICO on nuclear spins using the nuclear magnetic resonance system. We realize the ICO of two thermalizing channels to exhibit how the mechanism works, and show that the working substance can be cooled or heated albeit it undergoes thermal contacts with reservoirs of the same temperature. Moreover, we construct a single cycle of the ICO refrigerator based on the Maxwell's demon mechanism, and evaluate its performance by measuring the work consumption and the heat energy extracted from the low-temperature reservoir. Unlike classical refrigerators in which the coefficient of performance (COP) is perversely higher the closer the temperature of the high-temperature and low-temperature reservoirs are to each other, the ICO refrigerator's COP is always bounded to small values due to the nonunit success probability in projecting the ancillary qubit to the preferable subspace. To enhance the COP, we propose and experimentally demonstrate a general framework based on the density matrix exponentiation (DME) approach, as an extension to the ICO refrigeration. The COP is observed to be enhanced by more than 3 times with the DME approach. Our Letter demonstrates a new way for nonclassical heat exchange, and paves the way towards construction of quantum refrigerators on a quantum system.

Imaginary-time evolution using forward and backward real-time evolution with a single ancilla: First-quantized eigensolver algorithm for quantum chemistry

Imaginary-time evolution (ITE) on a quantum computer is a promising formalism for obtaining the ground state of a quantum system. The probabilistic ITE (PITE) exploits measurements to implement nonunitary operations, and it can avoid the restriction of dynamics to a low-dimensional subspace imposed by variational parameters unlike other types of ITE. In this paper, we propose a PITE approach that uses only one ancillary qubit. Unlike the existing PITE approaches, the one proposed here constructs, under a practical approximation, the circuit from forward and backward real-time evolution (RTE) gates as black boxes for the original Hamiltonian. Thus all efficient unitary algorithms for RTE can be transferred to the ITE without any modifications. Our approach can be used to obtain the Gibbs state at a finite temperature and partition function. We validate the approach via several illustrative systems where the trial states are found to converge rapidly to the ground states. In addition, we discuss its applicability to quantum chemistry by focusing on the scaling of computational cost; this leads to the development of a framework referred to as a first-quantized eigensolver. The nonvariational generic approach will expand the scope of practical quantum computation for versatile objectives.