is a t-bit approximation of , where t is the number of qubits in the clock register. Each phase estimation algorithm performs O(ms) measurements, resulting in a circuit of depth and size O(ms). 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 . Entanglement-free Heisenberg-limited phase estimation ... (PDF) Quantum computing, phase estimation and applications ... differential equations, etc. 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. 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. Entanglement-free Heisenberg-limited phase estimation | Nature First an ancilla register is prepared as |0i and then the transform xi (21) Preface This is a set of lecture notes on quantum algorithms. Usefulness of an enhanced Kitaev phase-estimation ... proach to quantum phase estimation [1,19,20]. Circuit diagrams for m-bit implementations of QPE using the two approaches described in this Letter: Kitaev's QPE (top) and IPE (bottom).The variable s guides the cumulative average of each circuit sampling up to its final amount determined by the total number of resources (measurements). Algorithms that use phase estimation as a subroutine Examples: computation of physical properties, applying inverse . Rev. PDF MASSACHUSETTS INSTITUE OF TECHNOLOGY Media Laboratory MAS ... 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 . The resulting circuits ~ Sampling random circuits vs . 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. Clifford+T gates or ${T,H}$ gates. 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. circuit and increasing kappropriately, you can e ciently obtain as many bits of pas desired, and thus, of ˚. Compared with other flavors of phase estimation such as Kitaev QPE . 2 II. U is an approximation to the simulated time evolution exp (i H). circuit of depth polynomial in the number of qubits. We propose a physical implementation of the protocol using the dispersive coupling between an ancilla transmon qubit and a cavity mode in circuit-QED. 4 Phase Estimation Phase Detection is actually a special case of a more general algorithm called Phase Estimation, due to Kitaev[Kit97]. Quantum phase estimation is one of the most important subroutines in quantum computation. The objective of the algorithm is the following: 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. Advances in precision measurement have consistently led to important . The quantum phase estimation algorithm performs the mapping where is an eigenvector of a unitary operator U with an unknown eigenvalue . 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. A 90, 062313 - Published 5 December 2014. Dive into the research topics of 'Encoding a qubit into a cavity mode in circuit QED using phase estimation'. 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 key example is the measurement of optical phase, used in length metrology and many other applications. 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. . Related work Simulating Hamiltonian dynamics [Berry, Childs, Cleve, Kothari, RS] Solovey-Kitaev to . and is comprised of a qubits. [30] Even implementations of phase estimation with only a single an-cillary qubit will be of foremost importance. Quantum simulation of the Sachdev-Ye-Kitaev model by asymmetric qubitization . It serves as a central building block for many quantum algorithms. An iterative scheme for quantum phase estimation (IPEA) is derived (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 . 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. By itself, the phase estimation algorithm is a solution to a rather . 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. phase of the wave function to oscillate rapidly across space). • Arthur runs the "poor man's phase estimation" circuit on e-iAt and . The notation follows that of Kitaev's QPE in Figure 2. • 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 The Bravyi-Kitaev Transformation: Properties and Applications Andrew Tranter,[a,b] . However, iterative QPE offers a baseline level of accuracy after just a single repetition. Encoding a qubit into a cavity mode in circuit QED using phase estimation 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 (translations) in phase space. 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 . The phase estimation algorithm is a quantum subroutine useful for finding the eigenvalue corresponding to an eigenvector u of some unitary operator. [Cleve & Watrous , 2000] Andrew J. Landahl, University of New Mexico . 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. A quantum Fourier transform and its application to a quantum algorithm for phase estimation is discussed. -sized circuit with depth. 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 QPE Algorithm Kitaev's algorithm for Phase Estimation is an algorithm with two forms. In the context of quantum simulation, this unitary is usually the time evolution operator \(e^{-iHt}\). To do this we will map an arbitrary quantum circuit with a constrained output and an unconstrained input register (i.e. More precisely, given a unitary matrix U {\displaystyle U} and a quantum s Earlier work [8] has proposed a scheme for quantum . This circuit is then mapped back to a low-degree simulating Hamiltonian, using the Feynman-Kitaev circuit to-Hamiltonian construction. For Kitaev 2002 phase estimation and fast phase estimation, s equals O(log(m)) and O(log*(m)), respectively. CS 294-2 Phase Estimation 3/5/07 Spring 2007 Lecture 12 In this lecture we will describe Kitaev's phase estimation algorithm, and use it to obtain an alternate derivation of a quantum factoring algorithm. B. L. Higgins Centre for Quantum Dynamics, Griffith University, . The lower bound for the probability to get a correct result in a single run of the algorithm has . 0.1 Phase Estimation Technique 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. (b) Qubitization circuit with one estimator qubit, featuring the three unitaries V ̂, S ̂, and G ̂, where only the latter is applied conditionally. =−= Initially introduced by Alexei Kitaev in 1995 and Seth Lloyd, Phys. Kitaev PE: =2 , =−1,…0and use the circuit with =0and =/2 The quantum phase estimator receives at least one ancillary qubit and a calculational state comprised of multiple qubits. The phase gate may apply random phases to the ancillary qubit, which is used as a control to the controlled unitary gate. Usefulness of an enhanced Kitaev phase-estimation algorithm in quantum metrology and computation Tomasz Kaftal and Rafał Demkowicz-Dobrzański Phys. Kitaev's phase estimation algorithm is a . 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 . Solution: Phase 1 A. Kitaev Given: Controlled- gates, . Faster phase estimation requires the minimal number of measurements with a log∗ factor of reduction when the required precision n is large. distributions by performing quantum phase estimation [7]. Kitaev developed a quantum computing circuit that outputs the binary digits for the estimate of a phase one by one . Given such a constraint k, we propose an approach for the phase estimation for an eigenphase of exactly n-bit precision. The circuit requires O(ms) ancilla qubits, one per measurement, plus a additional qubits. chaos in random quantum circuits [15] and chaos in AdS 2 holography [16]. This is an alternative to the phase estimation algorithm. Phase estimation Shifted Legendre symbol problem Simon Õs problem Sparse Hamiltonian simulation . and is comprised of a qubits. 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. We will now look at this same problem again, but using the QFT in a more sophisticated way: by Kitaev's phase estimation algorithm. We will also use this technique to design quantum circuits for computing the Quantum Fourier Transform modulo an arbitrary positive integer. 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 . One bit of phase per round. The gate set is represented by a discrete universal set, e.g. The standard QPE algorithm utilizes the complete version of the inverse QFT. A well known approach to solve the problem is to use Solovay-Kitaev (SK) algorithm. Bayes risk) after measuring E. of each iteration. Each phase estimation algorithm performs O(ms) measurements, resulting in a circuit of depth and size O(ms). Quantum Phase Estimation and Arbitrary Size Quantum Fourier Transforms . 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 Shor's algorithm¶. More precisely, given a unitary matrix and a quantum state such that , the algorithm estimates the value of However, the main problem is that Phase Estimation (partial list) Textbook PE: =2 , =−1,…0and use the circuit with adaptive phases (semi-classical implementation of Fourier Transform, 1999). and phase estimation [25, 26]. The conceptual circuit for Kitaev's phase estimation algorithm is shown in Fig. QPE comes in many variants, but a large subclass of these algorithms (e.g. Such Fourier-based approach can deliver a phase estimate with an arbi- Here it the theorem: Theorem 4.1. 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. More. Measurement underpins all quantitative science. Quantum phase estimation (QPE) is a key component for a wide range of applications, . 83, 5162 ( 1999 ) He is best known for introducing the quantum phase estimation algorithm and the concept of the topological quantum computer while working at . It is the starting point for many other algorithms and relies on the inverse quantum Fourier transform. 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. P6: Show that the three qubit gate Gde ned by the circuit: is universal for quantum computation whenever is irrational. 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). 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 . 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. We compare the circuit constructions for Kitaev's phase estimation algorithm and the fast phase estima- tion algorithm in Section VI. Quantum Phase Estimation Algorithm Is a quantum algorithm to estimate the phase (or eigenvalue) of an eigenvector of a unitary operator. More details can be found in references [1]. 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 , 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. Let j ibe an eigenvector of U, also given (in a sense) as a "black box". The cost is in terms of the number of elementary gates, not just the number of measurements. T erhal and D. Weigand JARA Institute for Quantum Information, R WTH A achen University, 52056 Aachen, Germany Reference [19] by Kitaev is commonly recognized as the origin of the Fourier-based approach to quantum phase estimation, while Refs. 1a. Well the output of the phase kickback circuit is 1 . What are the advantages, and disadvantages of AQC compared to the circuit model? Solovay-Kitaev algorithm： →{H,T } an arbitrary single-qubit gate CNOT + single-qubit gate → an arbitrary n-qubit unitary gate universal set {⇤(X ),H,T} Universal quantum computation X X X X X X X X For example a rotation for {|000i, |111i}: U has been proposed recently. 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. Say Project Questions: Together they form a unique fingerprint. However, before each iteration of this circuit, the choice of (M, )canbeclassicallycalculated so as to minimise the expected posterior variance (i.e. We investigate the cost of three phase estimation procedures that require only constant-precision phase shift operators. Apply the k-th sub-circuit on the qubits and the phase information will be phase-kicked back to the ancilla qubit. The algorithm yields, with K + 1 bits of precision, an estimate φest of a classical phase parameter φ, where eiφ. The circuit requires O(ms) ancilla qubits, one per measurement, plus a additional qubits. . Estimate a phase value on a system of two qubits through Iterative Phase Estimation (IPE) algorithm. Thus it is in-structive to compare the iterative PEA with Kitaev's PEA. QPE comes in many variants, but a large subclass of these algorithms (e.g. 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. The work involves the conversion of the target Hamiltonian to a quantum phase estimation circuit embedded in 1D. 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. Develop new quantum circuits where L !polylog(1= ): "high-precision" quantum algorithms . Entanglement-free Heisenberg-limited phase estimation. 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. The superscripts on the kets indicate the names of the registers which store the corresponding states. Approximate Quantum Fourier Transform (AQFT): Title: Encoding a Qubit into a Cavity Mode in Circuit-QED using Phase Estimation. The conceptual circuit for Kitaev's phase estimation algorithm is shown in Fig. The disadvan-tage of the standard phase estimation algorithm is the high degree of phase-shift operators required. 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 . It has been shown that the approximate quantum Fourier transform can be successfully used for the phase estimation instead of the full one. The next DSP block is the digital phase estimation, required to recover the signal's carrier phase.A widely used carrier phase recovery scheme for PSK signals (e.g. Thus, the entire process is a single circuit (dn=ke stages) that cannot be divided into parallel processes. 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. The arbitrary constant-precision . 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. 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. phase-shift operators, to QPE with approximate quantum Fourier transform (AQFT). An iterative scheme for quantum phase estimation (IPEA) is derived a QMA verifier) into the ground state of a local Hamiltonian in . This gate is known as the Deutsch gate. 11,12 In quantum computing, the Kitaev . 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 (c) The idea underlying. Let us now show that a quantum computer can efficiently simulate the period-finding machine. In addition to the required qubits for storing the corresponding eigenvector, suppose we have additional k qubits available. phase estimation algorithm requires the preparation of a guid-ing state. Let U be an unitary operation on RM, given to a quantum algorithm as a \black box\. Kitaev's procedure for this proceeds in two steps. Quantum Computing: Suppose I want to obtain a gate sequence representing a particular 1 qubit unitary matrix. On the other hand, if we have Rev. 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. We will then use the latter to describe Shor's quantum algorithm for the Discrete Log Problem. What are some promissing physical systems in which to implement AQC? [1,20] con- tain further description and analysis of the Kitaev approach. Since implementing exponentially small phase-shift operators is costly or . 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. . . Lett. Figure 1. The circuit of a generalized Kitaev's algorithm. With this algorithm, phase estimation can be done using a single photon at a time for sequentially estimating each bit of the phase. A quantum phase estimator may include at least one phase gate, at least one controlled unitary gate, and at least one measurement device. Here, we . 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 In this implementation, the algorithm which uses a single Unitary matrix for phase estimation is used. 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 guiding state is an input state to the algorithm, . Encoding a Qubit into a Cavity Mode in Circuit-QED using Phase Estimation B.M. between Kitaev's original approach and QPE with AQFT in terms of the degree of phase shift operators needed. Moreover, each bit has to be measured only once . Circuit for Rejection Filtering Phase Estimation (RFPE). 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. Abstract. scarce. This is Kitaev's original approach for quantum phase estimation. Application to phase estimation. 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. 1a. In the IPEA scheme, the bits of the phase are measured directly, without any need for classical postprocessing. The original paper by Farhi, Goldstone, Gutmann and Sipser provides a good starting point, and a web search will reveal a lot of follow up work. For Kitaev 2002 phase estimation and fast phase estimation, s equals O(log(m)) and O(log*(m)), respectively. • Approximate Quantum Fourier Transform (AQFT): We demonstrate 7.1 Phase Estimation Technique In this section, we define the . Kitaev's algorithm is a very efficient algorithm in terms of quantum execution. In this work we consider Kitaev's algorithm for quantum phase estimation. I tried this implementation for SK algorithm. (a) Kitaev's phase estimation circuit . It is primarily intended for graduate students who have already taken an introductory course on quantum information. 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. Note that for the IPE case (top right gray box) a resource includes the conditional reset of . It requires us to apply K+ 1 unitaries, These are called Feynman-Kitaev history states. The k-th sub-circuit on the qubits and the phase gate may apply random phases kitaev phase estimation circuit the simulated evolution! Ground state of a generalized Kitaev & # x27 ; s QPE in Figure 2 resulting in a circuit depth! 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