Modified Grover operator for amplitude estimation
Shumpei Uno, Yohichi Suzuki, Keigo Hisanaga, Rudy Raymond, Tomoki Tanaka, Tamiya Onodera, Naoki Yamamoto
aa r X i v : . [ qu a n t - ph ] O c t Modified Grover operator for quantum amplitude estimation
Shumpei Uno , Yohichi Suzuki , Keigo Hisanaga , Rudy Raymond , Tomoki Tanaka ,Tamiya Onodera , and Naoki Yamamoto ∗ ,1,3 Quantum Computing Center, Keio University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522, Japan Mizuho Information & Research Institute, Inc., 2-3 Kanda-Nishikicho, Chiyoda-ku, Tokyo, 101-8443, Japan Department of Applied Physics and Physico-Informatics, Keio University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522, Japan IBM Quantum, IBM Research-Tokyo, 19-21 Nihonbashi Hakozaki-cho, Chuo-ku, Tokyo, 103-8510, Japan Mitsubishi UFJ Financial Group, Inc. and MUFG Bank, Ltd., 2-7-1 Marunouchi, Chiyoda-ku, Tokyo 100-8388, Japan
Abstract
In this paper, we propose a quantum amplitude estimation method that uses a modified Grover operator andquadratically improves the estimation accuracy in the ideal case, as in the conventional one using the standardGrover operator. Under the depolarizing noise, the proposed method can outperform the conventional one inthe sense that it can in principle achieve the ultimate estimation accuracy characterized by the quantum Fisherinformation in the limit of a large number of qubits, while the conventional one cannot achieve the same value ofultimate accuracy. In general this superiority requires a sophisticated adaptive measurement, but we numericallydemonstrate that the proposed method can outperform the conventional one and approach to the ultimate accuracy,even with a simple non-adaptive measurement strategy.
Quantum computers embody a physically feasible computational model which may exceed the limit of conventionalcomputers, and have been researched vigorously for several decades. In particular, the amplitude estimation algo-rithm [1–6] has attracted a lot of attention as a fundamental subroutine of a wide range of application-oriented quantumalgorithms, such as the Monte Carlo integration [7–13] and machine learning tasks [14–20]. However, those quantumamplitude estimation algorithms are assumed to work on ideal quantum computers, and their performance on noisycomputers should be carefully investigated. In fact, evaluation as well as mitigation of existing quantum algorithmson noisy quantum computers has become a particularly important research subject, since real quantum computingdevices are publicly available. The amplitude estimation algorithm is of course one of those focused algorithms [21–23].The amplitude estimation is the problem of estimating the (unknown) amplitude of a certain fixed target state, inwhich the parameter we want to calculate is encoded. If we naively measure the state N times and use the measurementresults to construct an estimator, then it gives the estimate with the mean squared error scaling as O (1 / √ N ) due tothe central limit theorem. In contrast, the conventional amplitude estimation algorithms [1–6], in an ideal setting,quadratically improve the estimation error to O (1 /N ), using the Grover operator (or the amplitude amplificationoperator in a wider sense) associated with the target state. Here let us reconsider the role of target state. In theGrover search problem [24], the goal is to hit the unknown target state with high probability; but in the amplitudeestimation problem, the goal is not to get the target state, but is to obtain an information of the unknown amplitude.Hence there might be some other “desirable state” whose hitting probability can be used to construct a more preciseestimator on the original amplitude, than the conventional Grover-based one. In particular, following the discussionin the first paragraph, this paper explores a noise-tolerant state as such a desirable state.Therefore the questions we should answer are as follows. Is there actually a noise-tolerant state, such that its hittingprobability leads to a better estimator than the case using the target state, in the presence of noise? A candidateof such a noise-tolerant state would be a product state, e.g., | i n , which may be more robust than the (entangled)target state. Moreover, if it exists, how much does the new estimator reduce the estimation error? Does it achieve theoptimal precision under some condition? This paper provides affirmative answers to all these questions.The analysis is based on the 1-parameter quantum statistical estimation theory [25–28]. A one-sentence summaryof the theorem used in this paper is that, under some reasonable assumptions, the estimation error is generally lowerbounded by the inverse of classical Fisher information for a fixed measurement, which is further lower bounded bythe inverse of quantum Fisher information without respect to measurement. Hence the quantum Fisher informationgives the ultimate lower bound of estimation error, although constructing an estimator that achieves the quantumFisher information generally requires a sophisticated adaptive measurement strategy, which is difficult to implementin practice [29–34]. In this paper, we assume the depolarizing error, which was in fact identified as a minimal noisemodel observed in the real device (IBM Q) [23]. Then we take the product state | i n +1 as a noise-tolerant state andshow that, like the conventional amplitude amplification (Grover search) operator G , a newly introduced “transformed ∗ e-mail address: [email protected] Q realizes a 2-dimensional rotation of | i n +1 in the Hilbert space and eventuallyleads to alternative estimator. Then the Fisher information is calculated; F c ,G ( · ) and F q ,G ( · ) denote the classical andquantum Fisher information of the state generated by G , respectively; also F c ,Q ( · ) and F q ,Q ( · ) denote the classical andquantum Fisher information of the state generated by Q , respectively. Below we list the contribution of this paper. • In the ideal case, F c ,G ( N q ) = F q ,G ( N q ) = F c ,Q ( N q ) = F q ,Q ( N q ) = 4 N , where N q denotes the number ofoperations (note that they are all independent to the unknown parameter). That is, there is no differencebetween G - and Q -based methods, and in both cases there exists an estimator (e.g., the maximum likelihoodestimator) that achieves the ultimate precision. • Under the depolarizing noise, F c ,G ( N q ) | env ≤ F c ,Q ( N q ) | env ≤ F q ,G ( N q ) = F q ,Q ( N q ), where ·| env denote theupper envelope of the Fisher information as a function of N q . That is, the Q -based method using the noise-tolerant state is better than the conventional G -based method, in the sense of Fisher information. In particular, F c ,Q ( N q ) | env → F q ,Q ( N q ) holds and hence the ultimate precision is achievable in the limit of a large number ofqubit. • Despite the above-mentioned general difficulty in achieving the quantum Fisher information, we show numericallythat, under the depolarizing noise, a simple non-adaptive estimator for Q -based method achieves the precisionabout 1 / / G -based method, which is about 1 . . /F q ,Q ( N q ).The rest of this paper is organized as follows. In Sec. 2, we present the conventional G -based amplitude estimationmethod, and its classical and quantum Fisher information under the depolarizing noise. We also show there that the G -based method can never attain the quantum Fisher information under the noise. Next in Sec. 3, we propose analternative Q -based method using the noise-tolerant state, and discuss the classical and quantum Fisher information.Then it is shown that the quantum Fisher information can be achieved in the limit of a large number of qubits.In Sec. 4, we demonstrate by numerical simulation that the proposed Q -based method achieves a better estimationaccuracy than the G -based method, with the simple non-adaptive measurement strategy. Finally, we conclude thispaper in Sec. 5. The quantum amplitude estimation is the problem of estimating the value of an unknown parameter θ of the followingstate, given a unitary operator A acting on the ( n + 1)-qubit initial state | i n +1 : A | i n +1 = cos θ | ψ i n | i + sin θ | ψ i n | i , (2.1)where | ψ i n and | ψ i n are normalized n -qubit states. Note that the state (2.1) has an additional ancilla qubit torepresent solutions (“good” subspace), in contrast to the original Grover search which searches for solutions in n -qubitspace. Measuring the ancilla qubit N shot times gives us a simple estimate with estimation error (the root squaredmean error) scaling as 1 / √ N shot due to the central limit theorem. The purpose of amplitude estimation algorithms isto reduce the cost to achieve the same error; in this paper, the cost of algorithm is counted by the number of querieson the operator A and its inverse A † , as in the previous works [1–6]. This section is devoted to provide an overviewof this conventional quantum-enhanced method for both noiseless and depolarizing-noisy cases, with particular focuson the classical and quantum Fisher information. The discussion mainly follows the previous papers [2, 23, 35, 36]. The recent amplitude estimation algorithms [2–6], which do not require the expensive phase estimation, are composedof the following two steps. The first step is to amplify the amplitude of the target state and perform the measurementon the state, and the second step is to estimate the amplitude using the measurement results and a subsequent classicalpost-processing.The first step uses the amplitude amplification operator G defined as [1] G = AU A † U f , (2.2)where U and U f are the reflection operators defined as U = − I n +1 + 2 | i n +1 h | n +1 , (2.3) U f = − I n +1 + 2 I n ⊗ | i h | . (2.4)Here, I n is the identity operator on the n qubits space. Via m times application of the operator G on the state (2.1),we obtain G m A | i n +1 = cos ((2 m + 1) θ ) | ψ i n | i + sin ((2 m + 1) θ ) | ψ i n | i . (2.5)2ote now that the number of queries of A and A † is 2 m + 1. In the following, we regard this state as a (vector-valued)function of the general number of queries, N q , instead of m ; i.e., | ψ G ( θ, N q ) i ≡ cos ( N q θ ) | ψ i n | i + sin ( N q θ ) | ψ i n | i , (2.6)but remember that | ψ G ( θ, N q ) i can take only the odd number in N q . We then measure the state ρ G ( θ, N q ) = | ψ G ( θ, N q ) i h ψ G ( θ, N q ) | via the set of measurement operators E G, = I n ⊗ | i h | and E G, = I n ⊗ | i h | , whichdistinguishes whether the last bit is 0 or 1; the probability distribution p G ( i ; θ, N q ) = Tr( ρ G ( θ, N q ) E G,i ) is thencalculated as p G (0; θ, N q ) = cos ( N q θ ) ,p G (1; θ, N q ) = sin ( N q θ ) . (2.7)The second step is to post-process the measurement results sampled from the probability distribution (2.7) toestimate the parameter θ . If we have N shot independent measurement results, the mean squared error of any unbiasedestimate ˆ θ from the measurement results obeys the following Cram´er–Rao inequality: E h ( θ − ˆ θ ) i ≥ N shot F c ,G ( θ, N q ) , (2.8)where E [ · · · ] represents the expectation of measurement results with respect to the probability distribution (2.7). Theclassical Fisher information F c ,G ( θ, N q ) associated with the probability distribution (2.7) is given by F c ,G ( θ, N q ) = E "(cid:18) ∂∂θ ln p G ( i ; θ, N q ) (cid:19) = X i ∈{ , } (cid:18) ∂∂θ ln p G ( i ; θ, N q ) (cid:19) p G ( i ; θ, N q ) = 4 N q . (2.9)It is well known that the maximum likelihood estimation is able to asymptotically attain the lower bound of Cram´er–Rao inequality in the limit of a large number of samples, N shot → ∞ [37]. In the previous paper [2], it was demon-strated by a numerical simulation that the mean squared error obtained by the maximum likelihood estimation with N shot = 100 well agrees with the Cram´er–Rao lower bound. By combining the Fisher information (2.9) with theCram´er–Rao lower bound (2.8), we find that the root squared estimation error scales as 1 / ( N q √ N shot ). Denoting thetotal cost as n = N q N shot , the root squared error scales as 1 /n in the large N q limit, which is a quadratic improvementover the naive repeated measurement onto the initial state (2.1), whose error scales as 1 / √ N shot = 1 / √ n .Next, we calculate the quantum Fisher information for this amplitude estimation problem. Although we obtainthe classical Fisher information (2.9) with the fixed measurement operators, E G, and E G. , there is an innumerabledegree of freedom in choosing a set of measurement operators, and other measurements may provide larger classicalFisher information. However, it is known that any unbiased estimate ˆ θ resulting from an arbitrary measurement(including POVM) obeys the following quantum Cram´er–Rao inequality [25–27]: E h ( θ − ˆ θ ) i ≥ N shot F q ,G ( θ, N q ) , (2.10)where F q ,G ( θ, N q ) is the quantum Fisher information defined as F q ,G ( θ, N q ) = Tr (cid:16) ρ G ( θ, N q ) L S ( ρ G ( θ, N q )) (cid:17) . (2.11)Here L S ( ρ G ( θ, N q )) is the symmetric logarithmic derivative (SLD) , which is defined as the Hermitian operatorsatisfying ˙ ρ G ( θ, N q ) = 12 ( ρ G ( θ, N q ) L S ( ρ G ( θ, N q )) + L S ( ρ G ( θ, N q )) ρ G ( θ, N q )) . (2.12)Here the overdot represents the derivative with respect to θ . If the output state is a pure state, the quantum Fisherinformation is explicitly given by F q ,G ( θ, N q ) = 4 (cid:16) h ˙ ψ G ( θ, N q ) | ˙ ψ G ( θ, N q ) i − |h ˙ ψ G ( θ, N q ) | ψ G ( θ, N q ) i| (cid:17) = 4 N . (2.13)In our case, this is exactly the same as the classical Fisher information (2.9). This means that the measurementoperators E G, and E G, , which measure only the last qubit, is the optimal measurement in the absence of noise.It is worthwhile to comment that the amplitude estimation problem described above can be regarded as theparameter estimation problem embedded in the unitary operator G . Such a parameter estimation problem is one ofthe main target of the field of quantum metrology [39, 40]. In the quantum metrology, it is well known that the rootsquared error of the estimation for a single unitary operation on each of the N independent initial states is 1 / √ N ,which is called the standard quantum limit or simply the shot noise, while the entanglement or multiple sequentialoperation quadratically improves the estimation error to 1 /N , which is called the Heisenberg scaling. Unfortunately,it is also known that this quadratic improvement is very fragile and is ruined even with the infinitesimally smallnoise [41–43]. We encounter the same issue in the amplitude estimation problem as described below. There are other definitions of quantum Fisher information, but it is known that the SLD gives the tightest bound for one parameterestimation problem [26, 38]. .2 Depolarizing Noise Case Here, we describe a case under the depolarizing noise. As a noise model, we assume that the following depolarizingnoise acts on the system every time when operating A and A † : D ( ρ ) = rρ + (1 − r ) I n +1 d , (2.14)where ρ is an arbitrary density matrix, r is a known constant satisfying r ∈ (0 , d is the dimension of the( n + 1)-qubit system, i.e., d = 2 n +1 . Then, the amplitude amplification operation G m under the depolarizing process,with initial state A | i n +1 , gives ρ G,r ( θ, N q ) = r N q ρ G ( θ, N q ) + (1 − r N q ) I n +1 d , (2.15)where ρ G ( θ, N q ) = | ψ G ( θ, N q ) i h ψ G ( θ, N q ) | . Recall that ρ G,r ( θ, N q ) is represented as a function of general number ofqueries, N q , yet it takes N q = 2 m + 1. Measuring the state (2.15) using the set of measurement operators E G, and E G, , the probability distribution p G,r ( i ; θ, N q ) = Tr( ρ G,r ( θ, N q ) E G,i ) can be written as p G,r (0; θ, N q ) = r N q cos ( N q θ ) + 1 − r N q , (2.16) p G,r (1; θ, N q ) = r N q sin ( N q θ ) + 1 − r N q . (2.17)Note that this probability distribution is independent of the dimension of Hilbert space d . Then the classical Fisherinformation associated with this probability distribution is given as [23] F c ,G,r ( θ, N q ) = X i ∈{ , } (cid:18) ∂∂θ ln p G,r ( i ; θ, N q ) (cid:19) p G,r ( i ; θ, N q )= 4 N q2 sin ( N q θ ) cos ( N q θ ) r N q (cid:0) + (cid:0) cos ( N q θ ) − (cid:1) r N q (cid:1) (cid:0) + (cid:0) sin ( N q θ ) − (cid:1) r N q (cid:1) . (2.18)This classical Fisher information has the following upper envelope depicted in Fig. 1, which does not depend on theunknown parameter θ : F c ,G,r ( N q ) | env = 4 N q2 r N q . (2.19)From the Cram´er–Rao inequality, 1 /F c ,G,r ( N q ) | env gives a lower bound of the estimation error, which is (approxi-mately) attainable by employing an adaptive measurement strategy as proven in the channel estimation problem [28].The above expression shows that, even with a very small noise, the Heisenberg scaling is not achievable asymptoticallyin the large N q limit, due to the factor of r N q . Note that, nonetheless, it is possible to make a constant factorimprovement over the naive sampling onto the initial state (2.1), depending on the magnitude of noise. The Fisherinformation of the naive sampling, i.e., Eq. (2.19) with N q = 1, is about 4, while the Fisher information (2.19) for N q ≫ N q rather than the asymptotic behavior,which is the subject of the following.Next, like the ideal case, let us consider the quantum Fisher information. As in the previous work on the channelestimation problem [35, 36], exploiting the fact that the state of Eq.(2.15) can be represented in an exponential form,the SLD operator can be calculated as L S ( ρ G ( θ, N q , r )) = 2 r N q d + (cid:0) − d (cid:1) r N q ˙ ρ G ( θ, N q ) . (2.20)Then the quantum Fisher information can be given as F q ,G,r ( θ, N q ) = 4 N q2 r N q d + (cid:0) − d (cid:1) r N q . (2.21)Importantly, F q ,G,r ( θ, N q ) does not depend on the unknown parameter θ . This fact allows us to obtain F q ,G,r ( θ, N q ) ≥ F c ,G,r ( θ, N q ) | env ≥ F c ,G,r ( θ, N q ) for any value of θ . In the case of single qubit, n = 1 or equivalently d = 2, we have F q ,G,r ( θ, N q ) = F c ,G,r ( θ, N q ) | env . But F q ,G,r ( θ, N q ) > F c ,G,r ( θ, N q ) | env holds when n ≥
2. In particular, in the limitwhere the qubit number is infinitely large, it follows that F q ,G,r ( θ, N q ) = 4 N q2 r N q , (2.22)which is bigger than F c ,G,r ( θ, N q ) | env by the factor r N q . That is, the conventional G -based method presented in thissection cannot achieve the quantum Fisher information, except the case n = 1. In the next section, we present analternative method that can attain the quantum Fisher information in the limit of a large number of qubits.4
100 200 300 4000100020003000400050006000
Number of Queries N q F i s h e rI n f o r m a ti on F c , G , r H Θ= (cid:144) N q L F c , G , r H Θ= (cid:144) N q L F c , G , r H Θ= (cid:144) N q L F c , G , r H N q L env Figure 1:
Relationship between the number of queries and the classical Fisher information. The solid lines represent theclassical Fisher informations (2.18) with several values of θ (1 /
5, 1 /
10, and 1 / r = 0 . It is worthwhile to mention here about the input state of the above method. The input state A | i n +1 is an equallyweighted superposition of the two eigenstates of the amplitude amplification operator G , which is known as the optimalinput state in the sequential strategy of the channel estimation problem that sequentially applies the unitary G ontosingle probe [36, 44]. Therefore, the quantum Fisher information (2.21) is optimal for any method using the operator G . Furthermore, we prove in Appendix that the quantum Fisher information (2.21) is optimal when employing theoperator A . In this section, we propose an alternative amplitude estimation method, which provides a larger classical Fisherinformation than that of the conventional G -based method, under the depolarizing noise. As in the previous section,we first describe the noiseless case and then turn to the case under depolarizing noise. We first define the following modified amplitude amplification operator: Q = U A † U f A (3.1)This operator is very similar to the conventional amplitude amplification operator G = AU A † U f given in Eq. (2.2),but there is a big operational meaning as follows. Note that G induces a rotation by 2 θ in the space spanned by | ψ i n | i and | ψ i n | i , or equivalently A | i n +1 and | ψ i n | i . Then, because of Q = A † GA , we find that Q inducesa rotation by 2 θ in the space spanned by | i n +1 and A † | ψ i n | i (see Fig. 2). Thus, application of Q m on the initialstate | i n +1 produces | ψ Q ( θ, N q ) i := Q m | i n +1 = cos( N q θ ) | i n +1 + sin( N q θ ) | φ i n +1 , (3.2)where | φ i n +1 = A † ( − sin θ | ψ i n | i + cos θ | ψ i n | i )= 1sin 2 θ ( Q − cos 2 θ ) | i n +1 . (3.3)As in the G case, | ψ Q ( θ, N q ) i is represented as a (vector-valued) function of the general number of queries, N q , whichnow takes only the even numbers, i.e., N q = 2 m .In this method, we employ E Q, = | i n +1 h | n +1 and E Q, = I n +1 − E Q, as the set of measurement operators,which distinguishes whether all qubits are 0 or not. Measuring the state (3.2) via { E Q, , E Q, } yields the probabilitydistribution as p Q (0; θ, N q ) = cos ( N q θ ) ,p Q (1; θ, N q ) = sin ( N q θ ) . (3.4)5 igure 2: The graphical explanation of the action of G (left) and Q (right) on the initial state. The operator G defined inEq. (2.2) can be regarded as the rotation with angle 2 θ in the space spanned by | ψ i n | i and | ψ i n | i . First, theaction of U f reflects the initial state A | i n +1 with respect to | ψ i n | i , and then the action of AU A † reflects theresulting state with respect to the initial state A | i n +1 , so that the rotation with angle 2 θ is achieved. On the otherhand, the operator Q defined in Eq. (3.1) can be regarded as the rotation with angle 2 θ in the space spanned by | i n +1 and | φ i n +1 , i.e., linear transformation via A of the space spanned by | ψ i n | i and | ψ i n | i . In fact, first A † U f A reflects the initial state | i n +1 with respect to A † | ψ i n | i = cos θ | i n +1 − sin θ | φ i n +1 , and then U reflectsthe resulting state with respect to the initial state | i n +1 ; as a result, Q | i n +1 is a 2 θ -rotated state of | i n +1 . The classical Fisher information with respect to this probability distribution can be calculated as F c ,Q ( θ, N q ) = 4 N . (3.5)This indicates that the estimation error using the operator Q also obeys the Heisenberg scaling in the absence of noise.The quantum Fisher information for the state (3.2) can be calculated in the same manner as Eq. (2.13); F q ,Q ( θ, N q ) = 4 N . (3.6)The classical Fisher information (3.5) coincides with the quantum one (3.6), while they coincide with the Fisherinformation (2.9) and (2.13) for the G -based method. This indicates that the choice of the amplitude amplificationoperator G or Q does not cause any advantage/disadvantage in the noiseless case. We assume that the same depolarizing noise (2.14) introduced in Sec. 2.2 acts on the system every time when operating A . Then, the final state after m times operation of Q together with the noise process, for the initial state | i n +1 , isgiven by ρ Q,r ( θ, N q ) = r N q ρ Q ( θ, N q ) + (1 − r N q ) I n +1 d , (3.7)where ρ Q ( θ, N q ) = | ψ Q ( θ, N q ) i h ψ Q ( θ, N q ) | . Here, we again use the notation N q = 2 m . Measuring this state using theset of measurement operators E Q, and E Q, yields the probability distribution: p Q,r (0; θ, N q ) = r N q cos ( N q θ ) + (1 − r N q ) 1 d , (3.8) p Q,r (1; θ, N q ) = r N q sin ( N q θ ) + (1 − r N q ) d − d . (3.9)The classical Fisher information associated with this probability distribution is calculated as F c ,Q,r ( θ, N q ) = 4 N q2 sin ( N q θ ) cos ( N q θ ) r N q (cid:0) d + (cid:0) cos ( N q θ ) − d (cid:1) r N q (cid:1) (cid:0) d − d + (cid:0) sin ( N q θ ) − d − d (cid:1) r N q (cid:1) , (3.10)which has the following upper envelope that does not depend on the unknown parameter θ : F c ,Q,r ( N q ) | env = 4 N r N q + 8 N q2 d − d (cid:0) − r N q (cid:1) − N (cid:0) − r N q (cid:1) p ( d −
1) ( d − r N q ) (( d − r N q + 1) d . (3.11)Also the quantum Fisher information for the state (3.7) can be calculated as F q ,Q,r ( θ, N q ) = 4 N q2 r N q (cid:0) d + (cid:0) − d (cid:1) r N q (cid:1) . (3.12)6ike the case of G , this does not depend on θ , which leads that, because F c ,Q,r ( θ, N q ) | env is the maximization of F c ,Q,r ( θ, N q ) with respect to θ , we have F q ,Q,r ( θ, N q ) ≥ F c ,Q,r ( θ, N q ) | env ≥ F c ,Q,r ( θ, N q ).Now we compare the Fisher information of G and Q . The first notable fact is that their quantum Fisherinformation are identical, i.e., F q ,G,r ( θ, N q ) = F q ,Q,r ( θ, N q ). Hence a difference may appear for the values of classicalFisher information, especially their θ -independent envelopes F c ,Q,r ( N q ) | env and F c ,G,r ( N q ) | env . Actually, using theAM-GM inequality, we can prove F c ,Q,r ( N q ) | env ≥ N r N q + 8 N q2 d − d (cid:0) − r N q (cid:1) − N (cid:0) − r N q (cid:1) d ( (cid:0) d − r N q (cid:1) + ( d − (cid:0) ( d − r N q + 1 (cid:1) ) = 4 N q2 r N q = F c ,G,r ( N q ) | env . The equality holds only when d = 2. Summarizing, we have F c ,G,r ( N q ) | env ≤ F c ,Q,r ( N q ) | env ≤ F q ,Q,r ( θ, N q ) = F q ,G,r ( θ, N q ) . (3.13)Now considering the fact that F c ,G,r ( N q ) | env never achieve its ultimate bound F q ,G,r ( θ, N q ) except the case d = 2, weare concerned with the achievability in the Q case. Actually F c ,Q,r ( N q ) | env = F q ,Q,r ( θ, N q ) holds when d = 2. Butmore importantly, it can be proven thatlim d →∞ n F c ,Q,r ( N q ) | env − F q ,Q,r ( θ, N q ) o = 0 . This implies that, for a large number of qubits, the enveloped classical Fisher information nearly achieves the ultimatebound. The intuitive understanding for this remarkable difference between Q and G is that the pure noise effect forthe proposed Q -based method, the second term of Eq. (3.8), exponentially decreases with respect to the number ofqubits, which would allow us to efficiently extract the signal component given by the first term of Eq. (3.8). On theother hand, the pure noise effect on the G -based method, the second term of both of Eqs. (2.16) and (2.17) is about1 /
2, and thus the signal may be easily buried in the noise.Figure 3 illustrates the classical and quantum Fisher information of the G - and Q -based methods, as a function ofnumber of queries N q . Figure 3(a) shows that all the Fisher information presented above coincides with each other forthe single qubit case n = 1. For the case (b) n = 10, a clear difference between F c ,G,r ( N q ) | env and F c ,Q,r ( N q ) | env canbe seen; notably, the latter is already close to the ultimate bound F q ,Q,r ( N q ) even with such a small number of qubits.For the case (c) n = 100, F c ,Q,r ( N q ) | env almost reaches the bound F q ,Q,r ( N q ), while F c ,G,r ( N q ) | env does show nearlyzero change from that of (b). The shape of functions in (c) is almost the same as those shown in the case (d) n = ∞ ,where F c ,Q,r ( N q ) | env = F q ,Q,r ( N q ) takes the maximum value 16 / ( e ln r ) at N q = − / ln r , which is four times largerthan the maximum value of F c ,Q,r ( N q ) | env , which takes 4 / ( e ln r ) at N q = − / ln r .Therefore, as for the enveloped function, the Q -based method using the Q operator gives an estimator that notonly outperforms the G -based method but also, relatively easily, approaches to the ultimate estimation bound givenby the inverse of quantum Fisher information. Here we add a comment that the state | i n +1 is the optimal inputfor Q , in contrast to A | i n +1 being the optimal input for G mentioned in Sec. 2.2; see Appendix. This indicatesthat the quantum Fisher information obtained here is the optimal value for any method using the operator Q . Recallnow that, in general, to achieve the estimation accuracy given by the inverse of quantum Fisher information, weneed to elaborate a sophisticated adaptive measurement strategy, e.g., varying the number of amplitude amplificationoperations depending on the measurement result. In the next section, nonetheless, we demonstrate by numericalsimulations that the Q -based method has a solid advantage over the G -based method even when employing a simple,non-adaptive measurement method in which a sequence of amplitude amplification operators is scheduled in advance. The purpose of this simulation is to study the performance of an estimator constructed via the measurement result,compared to the inverse of Fisher information. Here we employ the same maximum likelihood estimator developedin the previous papers [2, 23], which is non-adaptive and thus is not guaranteed to achieve the θ -dependent quantumFisher information. This estimation method begins with preparing several states in parallel, where the amplitude ofthe target state is amplified according to a pre-scheduled sequence { m k } for k = 0 , , . . . ; then we make measurementson these states characterized by m k and finally combine all the measurement results to construct the maximumlikelihood estimator on the amplitude parameter. In particular, to achieve the Heisenberg scaling in the region wherethe noise has little effect on the measurement results, we employ an exponentially increasing sequence m k = ⌊ b k − ⌋ for k = 0 , , . . . , where b is some real number greater than 1; in this work, we take b = 6 /
5. Also, we fix the numberof measurement to N shot = 100 for all k . The number of qubit is n = 100 (i.e., d = 2 ) and the noise strength is r = 0 .
99 as in the case (c) in Fig. 3. The parameter values used in the numerical simulation are listed in Table 1.We repeated the same experiment 200 times to evaluate the root mean squared error of the estimate ˆ θ . Since the Note that while we compare here as if each method takes all integer values, the actual number of queries N q only takes even and oddvalues for the G -based and Q -based methods, respectively.
200 400 600 800 10000500010000150002000025000
Number of Queries N q F i s h e rI n f o r m a ti on F q , G , r H Θ , N q L = F q , Q , r H Θ , N q L F c , Q , r H N q L env F c , G , r H N q L env (a) n = 1 ( d = 2) Number of Queries N q F i s h e rI n f o r m a ti on F q , G , r H Θ , N q L = F q , Q , r H Θ , N q L F c , Q , r H N q L env F c , G , r H N q L env (b) n = 10 ( d = 2 ) Number of Queries N q F i s h e rI n f o r m a ti on F q , G , r H Θ , N q L = F q , Q , r H Θ , N q L F c , Q , r H N q L env F c , G , r H N q L env (c) n = 100 ( d = 2 ) Number of Queries N q F i s h e rI n f o r m a ti on F q , G , r H Θ , N q L = F q , Q , r H Θ , N q L F c , Q , r H N q L env F c , G , r H N q L env (d) n → ∞ ( d → ∞ ) Figure 3:
Relationship between the number of queries and the Fisher information, with several values of the number of qubits, n . The envelope of the classical Fisher information of the conventional G -based method, F c ,G,r ( N q ) | env defined inEq. (2.19), is depicted as the (red) dotted line, and the envelope of the classical Fisher information of the proposed Q -based method, F c ,Q,r ( N q ) | env defined in Eq. (3.11), is depicted as the (blue) dashed line. The quantum Fisherinformation F q ,Q,r ( θ, N q ) = F q ,G,r ( θ, N q ) defined in Eq. (3.12) and Eq. (2.21) is depicted as the (green) solid line.The noise parameter is set as r = 0 . Q -based method with m = 0 does not depend on the parameter θ , the measurement result obtained forthe case m = 0 is not used for the Q case, but note that it is used for the conventional G case. Also recall that thedepolarization process (2.14) acts on the state every time when A is operated.Figure 4 shows the relationship between the root mean squared estimation error and the total number of queriesdefined as N totq = P k N shot N q ( m k ), for several values of target amplitude a = sin θ . In each subfigure, the simulationresults using the maximum likelihood estimator described above are plotted by the circle and triangle points for the G and Q cases respectively. Also the three lines depict the quantum and classical Cram´er–Rao lower bounds givenby the inverse of Fisher information. We then see that, for all the cases from (a) to (f) of Fig. 4, the simulationresults well approximate the classical Cram´er–Rao lower bound (CCRB), as expected from the asymptotic efficiencyof maximum likelihood estimate. In particular, in the short range of N q where the influence from the noise is small, the N q -dependence of the estimation error of the Q -based method is almost identical with the G -based method. Moreover,in this region, they are also almost identical with the error given by the quantum Fisher information, all exhibiting theHeisenberg scaling. More precisely, the G -based method shows the Heisenberg scaling up to about N totq ∼ . × ,while the Q -based method does up to N totq ∼ × . As N q increases, the estimation errors saturate to constantvalues due to the influence of noise, where there is a difference between Q and G ; that is, the saturated error of theformer is about half the latter. Also, the saturated error of the G case is about 2 . Q is about 1 . . Q -based method, even without an adaptivesetting, has a practical advantage over the G -based one. Table 1:
List of parameters for the numerical simulation number of measurements N shot m k , ( k = 0 , , , . . . ) ⌊ (6 / k − ⌋ noise parameter r . d target values a = sin θ { / , / , / , / , / , / } In this paper, we proposed the amplitude estimation method exhibiting the better estimation accuracy than theconventional G -based method under the depolarizing noise. Firstly, we showed that there is no measurement strategyto achieve the ultimate estimation accuracy in the conventional G -based method, except the 1-qubit case. In contrast,the proposed Q -based method produces a larger classical Fisher information than the G -based one (in the sense ofan enveloped function) whenever the number of qubit is larger than one, and moreover, it achieves the ultimateestimation accuracy in the limit of large number of qubits. Note that, achieving the ultimate estimation accuracygenerally requires a sophisticated adaptive algorithm, but our numerical simulations confirmed that the estimator inthe Q -based method well approaches to the ultimate bound even with a simple non-adaptive strategy.In this paper, we have numerically confirmed the advantage of the Q -based method to the G -based one for anon-adaptive measurement strategy, but concrete adaptive algorithms to achieve the quantum Fisher information areopen for future research. In addition, although the depolarizing noise was analyzed in this study, it is also importantto analyze other noises such as dephasing noise and measurement noise. Also, in this paper, we assume that the noiseparameter, the parameter r in Eq. (2.14), is known, but it is also important to analyze the case where the parameteris unknown, as studied in the recent paper [23]. Acknowledgement
This work was supported by MEXT Quantum Leap Flagship Program Grant Number JPMXS0118067285 and JP-MXS0120319794 9 ç ç çç ç çççççç ç ç ççççç ç ççç ç çççç çç ççç ç çççç ò ò ò ò ò ò òò ò ò ò òò ò òò òòò ò òòò òò òò òò òòò ò òòòòò
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100 1000 10 ´ - Total Number of Queries N q tot E s ti m a ti on E rr o r ò Simulation H Q L ç Simulation H G L QCRBCCRB H Q L CCRB H G L (f) a = 1 / Figure 4:
Relationship between the total number of queries N totq and the root mean squared error. The simulation pa-rameters are listed in Table 1. The root mean square error obtained by the maximum likelihood estimationusing the conventional G -based and proposed Q -based methods are plotted as (red) circles and (blue) triangles,respectively. The lower bounds of the Cram´er–Rao inequality obtained from the quantum Fisher information F q ,Q,r ( θ, N q ) = F q ,G,r ( θ, N q ), the classical Fisher information of the Q -based method F c ,Q,r ( θ, N q ), and the classicalFisher information of the conventional method F c ,G,r ( θ, N q ) are shown by the (green) solid line, the (blue) brokenline, and the (red) dotted line, respectively. ppendix Proof of the optimality of the quantum Fisher information In the main text, we presented the quantum Fisher information in the absence of noise, Eq. (2.13) and Eq. (3.6), andthose under the depolarizing noise, Eq. (2.21) and Eq. (3.12). In this Appendix, we prove that these quantum Fisherinformation are optimal when employing a sequential strategy.The main theorem is as follows.
Theorem 1.
Let | ψ N i be a state defined in a d -dimensional Hilbert space as | ψ N i = U N V N ( θ ) U N − V N − ( θ ) · · · U V ( θ ) U V ( θ ) U | i . (A.1) where U i ( i = 0 , , . . . , N ) represents an arbitrary unitary operator independent of θ , and V i ( θ ) ( i = 1 , . . . , N ) is aunitary operator which is differentiable with respect to θ and whose derivative is non-expansive (that is, | ∂V i ( θ ) ∂θ | v i| ≤|| v i| holds for any vector | v i ). When we estimate θ from the measurement results sampled from the state | ψ N i , thequantum Fisher information F satisfies the following inequalities:1. F ≤ N in the absence of noise.2. F ≤ N ( Q Ni =1 r i ) d + ( − d ) Q Ni =1 r i under the depolarizing noise, where V i ( θ ) ( i = 1 , , . . . , N ) is subject to the depolarizingnoise of strength r i . The operators A and A † of Eq. (2.1) can be regarded as examples of V i ( θ ) in this theorem, if the action of ∂A∂θ onstates other than | i n +1 and | φ i n +1 (and ∂A † ∂θ on states other than | ψ i n | i and | ψ i n | i ) is non-expansive. Therefore,we can conclude that the quantum Fisher information obtained in the main text is optimal in the context of Theorem 1.In the following, we give the proof of Theorem 1, based on the following three Lemmas. Lemma 2. h ˙ ψ N | ψ N i is a pure imaginary number. Here the overdot represents the derivative with respect to θ .Proof. Differentiating both sides of h ψ N | ψ N i = 1 with θ gives the equation h ˙ ψ N | ψ N i = − h ˙ ψ N | ψ N i ∗ . Lemma 3.
The inequality h v | ˙ V i ( θ ) | v i + h v | ˙ V i ( θ ) | v i ∗ ≤ holds for any normalized vector | v i and | v i . The equalityholds only when h v | ˙ V i ( θ ) | v i = 1 .Proof. It is straightforward from the assumption that the operator ˙ V i ( θ ) is non-expansive. Lemma 4.
The norm || ˙ ψ N i| has an upper bound || ˙ ψ N i| ≤ N .Proof. We can expand the norm as || ˙ ψ N i| = N X i,j =1 h | n +1 U † V † ( θ ) · · · U † i − ˙ V † i ( θ ) U † i · · · V † N ( θ ) U † N U N V N ( θ ) · · · U j ˙ V j ( θ ) U j − · · · V ( θ ) U | i n +1 . (A.2)The right hand side of this equation takes the maximum value only when each term in the summation take the value 1,which is derived from Lemma 3 for i = j and from the assumption that V i ( θ ) is a non-expansive map for i = j .From these lemmas, the first part of the Theorem 1 (in the absence of noise) is proved as follows. Proof.
Recall that the quantum Fisher information for a pure state | ψ N i is obtained as F = 4 (cid:18) h ˙ ψ N | ˙ ψ N i − (cid:12)(cid:12)(cid:12) h ˙ ψ N | ψ N i (cid:12)(cid:12)(cid:12) (cid:19) . (A.3)The first term in the parenthesis of the right hand side is upper bounded by N from Lemma 4, and the second termis upper bounded by 0. Thus, we obtain the upper bound of quantum Fisher information as F ≤ N . The equalityholds only when h ˙ ψ N | ˙ ψ N i = N and h ˙ ψ N | ψ N i = 0.Also, the second part of the Theorem 1 (under the depolarizing noise) is proved as follows. Proof.
Assuming that the depolarizing noise with strength r i of the form (2.14) acts after the operation of V i ( θ ), thestate of Eq. (A.1) becomes ρ rN = ˜ rρ N + (1 − ˜ r ) I n +1 d . (A.4)Here we denote ρ N = | ψ N i h ψ N | and ˜ r = Q Ni =1 r i . Following the previous papers [35, 36], SLD operator of this statecan be written as L S = 2˜ r d + (cid:0) − d (cid:1) ˜ r ˙ ρ N . (A.5)11hen the quantum Fisher information can be calculated as F = Tr (cid:0) ρ rN L S (cid:1) = r d + (cid:0) − d (cid:1) ˜ r ! h ˜ r (cid:16) h ˙ ψ N | ψ N i + h ψ N | ˙ ψ N i + h ˙ ψ N | ˙ ψ N i + |h ψ N | ˙ ψ N i| (cid:17) + 1 − ˜ rd (cid:16) h ˙ ψ N | ψ N i + h ψ N | ˙ ψ N i + 2 h ˙ ψ N | ˙ ψ N i (cid:17)(cid:21) . (A.6)The first term in the bracket of the right hand side takes the maximum only when h ψ N | ˙ ψ N i = 0 and h ˙ ψ N | ˙ ψ N i = N hold from Lemmas 2 and 4. The second term also takes the maximum under the same conditions. Substituting theseconditions into Eq. 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