kitaev phase estimation circuit

Gottesman, Kitaev and Preskill have formulated a way of encoding a qubit into an oscillator such that the qubit is protected against small shifts (translations) in phase space. Kitaev's procedure for this proceeds in two steps. Compared with other flavors of phase estimation such as Kitaev QPE . chaos in random quantum circuits [15] and chaos in AdS 2 holography [16]. 11,12 In quantum computing, the Kitaev . [1,20] con- tain further description and analysis of the Kitaev approach. Quantum Algorithms For: Quantum Phase Estimation ... Encoding a Qubit into a Cavity Mode in Circuit-QED using ... A guiding state is an input state to the algorithm, . Encoding a Qubit into a Cavity Mode in Circuit-QED using ... By itself, the phase estimation algorithm is a solution to a rather . Well the output of the phase kickback circuit is 1 . I tried this implementation for SK algorithm. The notation follows that of Kitaev's QPE in Figure 2. 0.1 Phase Estimation Technique Quantum simulation of the Sachdev-Ye-Kitaev model by asymmetric qubitization . Earlier work [8] has proposed a scheme for quantum . This is Kitaev's original approach for quantum phase estimation. Thus, the entire process is a single circuit (dn=ke stages) that cannot be divided into parallel processes. Together they form a unique fingerprint. Deterministic Encoding of a Qubit in a Cavity Mode Using ... The cost is in terms of the number of elementary gates, not just the number of measurements. The circuit of a generalized Kitaev's algorithm. The standard QPE algorithm utilizes the complete version of the inverse QFT. Title: Encoding a Qubit into a Cavity Mode in Circuit-QED using Phase Estimation. Dive into the research topics of 'Encoding a qubit into a cavity mode in circuit QED using phase estimation'. The arbitrary constant-precision . We analyze the use of phase shifts that simplify the estimation of successive bits in the estimation of unknown phase , By using increasingly accurate shifts we reduce the number of measurements to the point where only a single measurement is needed for each additional bit. A quantum phase estimation algorithm allows us to perform full configuration interaction (full-CI) calculations on quantum computers with polynomial costs against the system size under study, but it requires quantum simulation of the time evolution of the wave function conditional on an ancillary qubit, which makes the algorithm implementation on real quantum devices difficult. We will also use this technique to design quantum circuits for computing the Quantum Fourier Transform modulo an arbitrary positive integer. I hope this approach to solve the QPE problem made you ponder about how simple classical processing may sometimes pose an equivalent. Kitaev's Phase Estimation — QPE algorithms | Quantum Untangled Recently, the faster phase estimation (FPE) algorithm [ 11 ] shows FPE has a $\log *$ Footnote 2 factor of reduction in terms of the total number of . Develop new quantum circuits where L !polylog(1= ): "high-precision" quantum algorithms . Entanglement-free Heisenberg-limited phase estimation | Nature More details can be found in references [1]. A quantum phase estimator may include at least one phase gate, at least one controlled unitary gate, and at least one measurement device. (b) Qubitization circuit with one estimator qubit, featuring the three unitaries V ̂, S ̂, and G ̂, where only the latter is applied conditionally. The idea underlying. PDF A Generalised Variational Quantum Eigensolver Quantum phase estimation (QPE) is a key component for a wide range of applications, . Usefulness of an enhanced Kitaev phase-estimation algorithm in quantum metrology and computation Tomasz Kaftal and Rafał Demkowicz-Dobrzański Phys. Research Code Bayes risk) after measuring E. of each iteration. Encoding a qubit into a cavity mode in circuit QED using ... Approximate Quantum Fourier Transform (AQFT): phase-shift operators, to QPE with approximate quantum Fourier transform (AQFT). Related work Simulating Hamiltonian dynamics [Berry, Childs, Cleve, Kothari, RS] Solovey-Kitaev to . Application to phase estimation. In this thesis, attention is paid to small experimental testbed applications with respect to the quantum phase estimation algorithm, the core approach for finding energy eigenvalues. Apply the k-th sub-circuit on the qubits and the phase information will be phase-kicked back to the ancilla qubit. As in the case of the Deutsch-Jozsa algorithm, we shall exploit quantum parallelism and constructive interference to determine whether a complicated function has a certain global property that cannot be learned by evaluating the function only at a few points. We compare the circuit constructions for Kitaev's phase estimation algorithm and the fast phase estima- tion algorithm in Section VI. A quantum Fourier transform and its application to a quantum algorithm for phase estimation is discussed. We analyze the performance of repeated and adaptive phase estimation as the experimentally most viable schemes given a realistic upper limit on the number of photons in the oscillator. Circuit for Rejection Filtering Phase Estimation (RFPE). Algorithms that use phase estimation as a subroutine Examples: computation of physical properties, applying inverse . The conceptual circuit for Kitaev's phase estimation algorithm is shown in Fig. Alexei Yurievich Kitaev (Russian: Алексей Юрьевич Китаев; born August 26, 1963) is a Russian-American professor of physics at the California Institute of Technology and permanent member of the Kavli Institute for Theoretical Physics. the semi-classical version of textbook phase estimation [3, 4], Kitaev's phase estimation , Heisenberg-optimized versions ), are executed in an iterative sequential form using controlled-U k gates with a single ancilla qubit [7, 8] (see figure 1), or by direct . Phase Estimation (partial list) Textbook PE: =2 , =−1,…0and use the circuit with adaptive phases (semi-classical implementation of Fourier Transform, 1999). Preface This is a set of lecture notes on quantum algorithms. Let j ibe an eigenvector of U, also given (in a sense) as a "black box". An iterative scheme for quantum phase estimation (IPEA) is derived The phase gate may apply random phases to the ancillary qubit, which is used as a control to the controlled unitary gate. • Kitaev shows how to take this circuit and produce a Hamiltonian with the property that: • In the "yes case", the Hamiltonian's minimum eigenvalue is less than some quantity involving the This is an alternative to the phase estimation algorithm. Each phase estimation algorithm performs O(ms) measurements, resulting in a circuit of depth and size O(ms). Such Fourier-based approach can deliver a phase estimate with an arbi- is a t-bit approximation of , where t is the number of qubits in the clock register. Kitaev's QPE Algorithm Kitaev's algorithm for Phase Estimation is an algorithm with two forms. The resulting circuits ~ Sampling random circuits vs . What are the advantages, and disadvantages of AQC compared to the circuit model? Quantum Phase Estimation and Arbitrary Size Quantum Fourier Transforms . The work involves the conversion of the target Hamiltonian to a quantum phase estimation circuit embedded in 1D. More precisely, given a unitary matrix U {\displaystyle U} and a quantum s Encoding a Qubit into a Cavity Mode in Circuit-QED using Phase Estimation B.M. Project Questions: • Arthur runs the "poor man's phase estimation" circuit on e-iAt and . The circuit requires O(ms) ancilla qubits, one per measurement, plus a additional qubits. One bit of phase per round. =−= Initially introduced by Alexei Kitaev in 1995 and Seth Lloyd, Phys. (Kitaev's algorithm) Consider the quantum circuit where |u) is an eigenstate of U with eigenvalue Show that the top qubit is measured to be 0 with probability p = cos 2 (πϕ).Since the state |u) is unaffected by the circuit it may be reused; if U can be replaced by U k, where k is an arbitrary integer under your control, show that by repeating this circuit and increasing k appropriately, you . Quantum Phase Estimation Algorithm Is a quantum algorithm to estimate the phase (or eigenvalue) of an eigenvector of a unitary operator. He is best known for introducing the quantum phase estimation algorithm and the concept of the topological quantum computer while working at . Say circuit of depth polynomial in the number of qubits. scarce. Quantum Phase Estimation which cover a spectrum of possible methods: • Kitaev Hadamard Tests (KHT): The approach orig-inally proposed by Kitaev [17] relies on a pre-determined number of trials to achieve a desired target for the error-rate and precision of estimation. The lower bound for the probability to get a correct result in a single run of the algorithm has . differential equations, etc. Estimate a phase value on a system of two qubits through Iterative Phase Estimation (IPE) algorithm. Kitaev PE: =2 , =−1,…0and use the circuit with =0and =/2 APER Adaptive Phase Estimation by Repetition BCH formula Baker-Campbell-Hausdor Formula CDF Cumulative Distribution Function CNOT gate Controlled NOT Gate EPR Einstein-Podolsky-Rosen FPGA Field-Programmable Gate Array GKP code Code Proposed by Gottesman, Kitaev and Preskill IPEA Iterative Phase Estimation Algorithm RAM Random-Access Memory Authors: B. M. Terhal, D. Weigand (Submitted on 16 Jun 2015 , last revised 27 Nov 2015 (this version, v4)) Abstract: Gottesman, Kitaev and Preskill have formulated a way of encoding a qubit into an oscillator such that the qubit is protected against small shifts . Rev. However, before each iteration of this circuit, the choice of (M, )canbeclassicallycalculated so as to minimise the expected posterior variance (i.e. Phase estimation Shifted Legendre symbol problem Simon Õs problem Sparse Hamiltonian simulation . 2 II. Three models of computa- tion are discussed: the rst is a sequential model with limited parallelism, the second is a highly parallel model, and the third is a model based on a cluster of quantum computers. The disadvan-tage of the standard phase estimation algorithm is the high degree of phase-shift operators required. The algorithm yields, with K+ 1 bits of precision, an estimate ˚ est of a classical phase parameter ˚, where ei˚ is an eigenvalue of a uni-tary operator U. Phase estimation Last time we saw how the quantum Fourier transform made it possible to find the period of a function by repeated measurements and the greatest common divisor (GCD) algorithm. However, the main problem is that We provide an estimate that in a current experimental set-up one can prepare a good code state from a squeezed vacuum state using $8$ rounds of adapative phase estimation, lasting in . In the context of quantum simulation, this unitary is usually the time evolution operator $e^{-iHt}$. Due to the great difficulty in scalability, quantum computers are limited in the number of qubits during the early stages of the quantum computing regime. This gate is known as the Deutsch gate. -sized circuit with depth. However, iterative QPE offers a baseline level of accuracy after just a single repetition. a QMA verifier) into the ground state of a local Hamiltonian in . Phase Estimation In this lecture we will describe Kitaev's phase estimation algorithm, and use it to obtain an alternate derivation of a quantum factoring algorithm. It serves as a central building block for many quantum algorithms. Here, we . the semi-classical version of textbook phase estimation [3, 4], Kitaev's phase estimation , Heisenberg-optimized versions ), are executed in an iterative sequential form using controlled-U k gates with a single ancilla qubit [7, 8] (see figure 1), or by direct . P6: Show that the three qubit gate Gde ned by the circuit: is universal for quantum computation whenever is irrational. Phase estimation Last time we saw how the quantum Fourier transform made it possible to ﬁnd the period of a function by repeated measurements and the greatest common divisor (GCD) algorithm. There are two major classes of phase estimation algorithms, one suggested early on by Kitaev 10 and a second originating from the quantum Fourier transform. We will now look at this same problem again, but using the QFT in a more sophisticated way: by Kitaev's phase estimation algorithm. What are some promissing physical systems in which to implement AQC? Let U be an unitary operation on RM, given to a quantum algorithm as a \black box\. QPE comes in many variants, but a large subclass of these algorithms (e.g. Quantum Phase Estimation which cover a spectrum of possible methods: Kitaev Hadamard Tests (KHT): The approach orig-inally proposed by Kitaev [17] relies on a pre-determined number of trials to achieve a desired target for the error-rate and precision of estimation. The traditional Kitaev QPE protocol for estimating the eigenvalues relies on a number of repetitions of a quantum circuit, where more repetitions results in additional accuracy (and fewer repetitions means less accuracy). phase estimation algorithm requires the preparation of a guid-ing state. B. L. Higgins Centre for Quantum Dynamics, Griffith University, . Under such an assumption, for approaches that require repetitions, such as Kitaev's7 and others,9 parallelization cannot be done and the circuit depth is the same as the size of the circuit. It is primarily intended for graduate students who have already taken an introductory course on quantum information. Kitaev's algorithm is a very efficient algorithm in terms of quantum execution. The gate set is represented by a discrete universal set, e.g. We provide an estimate that in a current experimental set-up one can prepare a good code state from a squeezed vacuum state using $8$ rounds of adapative phase estimation, lasting in . phase of the wave function to oscillate rapidly across space). We will now look at this same problem again, but using the QFT in a more sophisticated way: by Kitaev's phase estimation algorithm. In quantum computing, the quantum phase estimation algorithm (also referred to as quantum eigenvalue estimation algorithm), is a quantum algorithm to estimate the phase (or eigenvalue) of an eigenvector of a unitary operator. proach to quantum phase estimation [1,19,20]. [30] Even We will also use this technique to design quantum circuits for computing the Quantum Fourier Transform modulo an arbitrary positive integer. • Approximate Quantum Fourier Transform (AQFT): Phase estimation is a procedure that, given access to a controlled unitary and one of its eigenvectors, estimates the phase of the eigenvalue corresponding to that eigenvector. In quantum computing, the quantum phase estimation algorithm (also referred to as quantum eigenvalue estimation algorithm ), is a quantum algorithm to estimate the phase (or eigenvalue) of an eigenvector of a unitary operator. Quantum phase estimation is one of the most important subroutines in quantum computation. In this thesis, attention is paid to small experimental testbed applications with respect to the quantum phase estimation algorithm, the core approach for finding energy eigenvalues. The phase estimation algorithm is a quantum subroutine useful for finding the eigenvalue corresponding to an eigenvector u of some unitary operator. QPE comes in many variants, but a large subclass of these algorithms (e.g. Entanglement-free Heisenberg-limited phase estimation. Faster phase estimation requires the minimal number of measurements with a log∗ factor of reduction when the required precision n is large. In this paper our goal is to ll this gap and introduce a more general phase estimation algorithm such that it is possible to realize a phase estimation algorithm with any degree of phase shift operators in hand. In addition to the required qubits for storing the corresponding eigenvector, suppose we have additional k qubits available. Having gone through previous labs, you should have noticed that the "length" of a quantum circuit is the primary factor when determining the magnitude of the errors in the resulting output distribution; quantum circuits with greater depth have . This circuit is then mapped back to a low-degree simulating Hamiltonian, using the Feynman-Kitaev circuit to-Hamiltonian construction. modi ed phase estimation procedures, the Kitaev- and the semiclassical Fourier-transform algo- rithms, using an arti cial atom realized with a superconducting transmon circuit. and phase estimation [25, 26]. Shor's algorithm¶. The objective of the algorithm is the following: and is comprised of a qubits. An iterative scheme for quantum phase estimation (IPEA) is derived Quantum Computing: Suppose I want to obtain a gate sequence representing a particular 1 qubit unitary matrix. T erhal and D. Weigand JARA Institute for Quantum Information, R WTH A achen University, 52056 Aachen, Germany between Kitaev's original approach and QPE with AQFT in terms of the degree of phase shift operators needed. For Kitaev 2002 phase estimation and fast phase estimation, s equals O(log(m)) and O(log*(m)), respectively. Here it the theorem: Theorem 4.1. has been proposed recently. (c) 83, 5162 ( 1999 ) In this lecture, we will see how to use the phase estimation circuit to perform factoring (Kitaev's algorithm)and Quantum Fourier Transform modulo an arbitrary positive integer. A 90, 062313 - Published 5 December 2014. The conceptual circuit for Kitaev's phase estimation algorithm is shown in Fig. Note that for the IPE case (top right gray box) a resource includes the conditional reset of . In this implementation, the algorithm which uses a single Unitary matrix for phase estimation is used. Abstract. In this work we consider Kitaev's algorithm for quantum phase estimation. In , it is shown that its circuit depth for QFT $^\dagger $ is about $1/14$ of that in Kitaev's approach when the constant-precision phase shift operator is precise to the third degree. U is an approximation to the simulated time evolution exp (i H). A key example is the measurement of optical phase, used in length metrology and many other applications. We demonstrate We propose a physical implementation of the protocol using the dispersive coupling between an ancilla transmon qubit and a cavity mode in circuit-QED. Rev. These are called Feynman-Kitaev history states. and is comprised of a qubits. Figure 1. With this algorithm, phase estimation can be done using a single photon at a time for sequentially estimating each bit of the phase. The circuit for Kitaev phase estimation is given as: By varying $\theta$, we are able to determine $\sin(2 \pi M \phi_k)$ and $\cos (2 \pi M \phi_k)$ from sampling the circuit and calculating the [Cleve & Watrous , 2000] Andrew J. Landahl, University of New Mexico . Clifford+T gates or ${T,H}$ gates. The superscripts on the kets indicate the names of the registers which store the corresponding states. A well known approach to solve the problem is to use Solovay-Kitaev (SK) algorithm. . More. The Precise Succinct Hamiltonian Problem •Definition: "Succinct Encoding" •We say a classical Turing machine M is a Succinct Encoding for 2k(n)x 2k(n)matrix A if: •On inputi∈{0,1}k(n), Moutputs non-zero elements in i-throw of A •Using at most poly(n)time and k(n)space •k(n)-Precise Succinct Hamiltonianproblem•Input: Succinct Encoding of 2k(n)x 2k(n)Hermitian PSD matrix A (a) Kitaev's phase estimation circuit . There are two major classes of phase estimation algorithms, one suggested early on by Kitaev10 and a second originating from the quantum Fourier transform.11,12 In quantum computing, the Kitaev algorithm was run as part of Shor's factorization algorithm13 and the Fourier transform algorithm was used in optics to measure for BPSK, QPSK, and 8PSK) is the feed-forward Mth power phase estimation [76] (or Viterbi and Viterbi algorithm [77]), the latter was used in the analysis of different transmission systems, described in the next part of the chapter. To do this we will map an arbitrary quantum circuit with a constrained output and an unconstrained input register (i.e. Kitaev's phase estimation algorithm is a . Since implementing exponentially small phase-shift operators is costly or . The circuit requires O(ms) ancilla qubits, one per measurement, plus a additional qubits. 1a. implementations of phase estimation with only a single an-cillary qubit will be of foremost importance. Gottesman, Kitaev, and Preskill have formulated a way of encoding a qubit into an oscillator such that the qubit is protected against small shifts (translations) in phase space. Each phase estimation algorithm performs O(ms) measurements, resulting in a circuit of depth and size O(ms). In the IPEA scheme, the bits of the phase are measured directly, without any need for classical postprocessing. 1a. Measurement underpins all quantitative science. We propose a detailed physical implementation of this protocol using the dispersive coupling between a transmon ancilla qubit and a cavity mode in circuit-QED. Reference [19] by Kitaev is commonly recognized as the origin of the Fourier-based approach to quantum phase estimation, while Refs. distributions by performing quantum phase estimation [7]. It has been shown that the approximate quantum Fourier transform can be successfully used for the phase estimation instead of the full one.