Quantum algorithm for universal implementation of projective measurement of energy
QQuantum algorithm for universal implementation of projective measurement of energy
Shojun Nakayama, Akihito Soeda,
1, 2 and Mio Murao
1, 3 Department of Physics, Graduate School of Science,University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, Japan Centre for Quantum Technologies, National University of Singapore, Singapore Institute for Nano Quantum Information Electronics,University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo, Japan
A projective measurement of energy (PME) on a quantum system is a quantum measurement,determined by the Hamiltonian of the system. PME protocols exist when the Hamiltonian is givenin advance. Unknown Hamiltonians can be identified by quantum tomography, but the time costto achieve a given accuracy increases exponentially with the size of the quantum system. In thisletter, we improve the time cost by adapting quantum phase estimation, an algorithm designedfor computational problems, to measurements on physical systems. We present a PME protocolwithout quantum tomography for Hamiltonians whose dimension and energy scale are given butotherwise unknown. Our protocol implements a PME to arbitrary accuracy without any dimensiondependence on its time cost. We also show that another computational quantum algorithm may beused for efficient estimation of the energy scale. These algorithms show that computational quantumalgorithms have applications beyond their original context with suitable modifications.
PACS numbers: 03.67.-a, 03.67.Ac, 06.20.Dk
INTRODUCTION
Projective measurement of energy (PME) is a quantumcounterpart of an ideal energy measurement in classicalmechanics. A PME on a given system sets the system toan energy eigenstate and returns the corresponding en-ergy eigenvalue. A PME alone has no effect on a systemalready in an energy eigenstate, thus can be used to con-firm that the system remains in the initial energy eigen-state by repeating the same PME and observing thatthe outcomes remain unchanged. These properties makePME suitable for detecting small effects on a quantumsystem that is subject to an external influence such asgravity wave [1] or thermal fluctuation [2–4].In practice, a device that implements a quantum mea-surement must include a destructive component such asa photon detector. PME being a nondestructive mea-surement requires another quantum system as a “probe”.The system (commonly referred to as “target”) interactswith the probe, and a direct measurement is performedonly on the probe after the interaction (Fig. 1).An implementation protocol of PME is known for sys-tems whose Hamiltonian H is given in advance [5]. Theprotocol chooses the interaction between the probe andtarget according to H , so that the two quantum systemsare appropriately entangled. The entanglement assuresthat the measurement on the probe sets the target toan energy eigenstate, and that the outcome of the mea-surement identifies the respective energy eigenvalue. Thetime needed to induce the entanglement can be madearbitrarily short by increasing the strength of the inter-action. Thus, PME of known H can be implementedinstantaneously in principle.This protocol, however, does not take into account the FIG. 1. (Color online) Schematic diagrams of PME protocolson a system of unknown self-Hamiltonian H (labeled “Tar-get”). The blue boxes exp( − iHt ) denote the target beinglet evolve for time t with (cid:126) = 1. M is a quantum measure-ment which returns a numerical outcome. The implementa-tion time is lower-bounded by the time required to inducethe evolution of the target system, since there is no limit tothe strength of the interaction induced from outside on thesystem in principle. In the top protocol, H is identified byquantum (process) tomography, with at least N QT = O ( d )uses of the time evolution exp( − iHt ) for d -dimensional sys-tems. The quantum algorithmic PME (bottom) proposed inthis letter avoids quantum tomography and all interactionsare H -independent. time required to identify H . Let us estimate the timecost by analyzing quantum process tomography [6, 7] onthe time evolution of the system. Process tomographyinvolves setting the target to various “test states” andmeasuring the expectation value of appropriate observ-ables for each resulting state after the time evolution. Acomplete process tomography for a system described bya d -dimensional Hilbert space H = C d requires a number a r X i v : . [ qu a n t - ph ] A p r of observable, O ( d ), equal to the number of parametersin the Hamiltonian [8].An accurate estimation of the expectation values needsto accumulate sufficient statistics. Each use of the timeevolution costs time t , hence the total time cost for the to-mography to achieve a given accuracy for a d -dimensionalsystem scales at least O ( d ). This implies that, if H isunknown, the total implementation time for PME viaprocess tomography grows at least exponentially in thenumber of subsystems due to the exponential growth ofthe total dimension for composite systems.Tomography is required even if a PME is to be per-formed only once. It extracts enough information toidentify all the eigenspaces and eigenvalues of H , so thedimension dependence is unavoidable. A single use ofPME, however, does not reveal the exact description ofthe energy eigenspaces or the whole energy spectrum. Amore efficient PME protocol is needed.To improve a PME protocol is to find a better quantumalgorithm. Some quantum algorithms are known to pro-vide an efficient solution to computational problems [9].These algorithms, however, assume that the dynamics ofa quantum system can be “switched off” at will, whichdoes not hold in this problem.In this paper, we introduce a more efficient PMEprotocol and show that we can remove the dimension-dependence in the time cost, for unknown Hamiltonianswhose energy scale is given. Our protocol exploits a mod-ified version of quantum phase estimation (QPE) [10]. Fi-nally, we discuss an estimation protocol for the energyscale, based on an estimation of the trace of a unitary op-erator. We will show that another computational quan-tum algorithm, adapted from Ref. [11], performs moreefficiently than a complete tomography. PROJECTIVE MEASUREMENT BY QPE
QPE is designed so that each run returns a good es-timate for some eigenvalue of a given unitary operator U = (cid:80) dk =1 exp( iθ k ) | θ k (cid:105)(cid:104) θ k | on H = C d . Note that weassume 0 ≤ θ k < π . For a given input state | θ k (cid:105) , thecorresponding phase θ k is estimated by QPE.An essential building block of QPE is a controlled-unitary operation C U , which is a unitary gate that con-ditionally operates U on a d -dimensional target systemdenoted by H t = C d according to the state of an extra control qubit denoted by H c = C . Formally, the actionof C U on H c ⊗ H t is defined by C U | (cid:105)| ϕ (cid:105) = | (cid:105)| ϕ (cid:105) and C U | (cid:105)| ϕ (cid:105) = | (cid:105) U | ϕ (cid:105) for any | ϕ (cid:105) ∈ H t where {| (cid:105) , | (cid:105)} forms the computational basis of H c . To achieve N -bits estimation, QPE uses N control qubits for apply-ing ( C U ) l − between the l -th control qubit for each l ∈ { , . . . , N } and the target. We obtain an N -bit string { n , · · · , n N } of outcomes by the final measurements onthe N control qubits in the computational basis. By defining n N := (cid:80) Nl =1 l − n l and f ( n N ) := n N / N , thephase θ k is estimated as θ k = 2 πf ( n N ).In the limit N → ∞ , f ( n N ) can be regarded as acontinuous variable f with 0 ≤ f ≤
1. For any θ k ,the probability p N ( f ( n N ) | θ k ) to obtain n N for an ini-tial state | θ k (cid:105) approaches the delta function δ ( f − θ k / π )in distribution. The distance between p N ( f ( n N ) | θ k ) and δ ( f − θ k / π ) is independent of d . At the same limit, thetarget is transformed to an eigenstate by a projectiononto the corresponding eigenspace induced by the finalmeasurements of QPE. Interested readers may refer toAppendix. for details of QPE. QPE AND UNIVERSAL CONTROLLIZATION
The evolution of a target with Hamiltonian H for time t is given by the unitary operator U ( t ) = exp( − iHt ),with (cid:126) = 1. It may appear that QP on U ( t ) readily im-plements a projection onto the eigenspace correspondingto the estimated phase of U ( t ), which is also the desiredPME of H up to ambiguity due to the phase periodic-ity. QPE assumes that U is available in its quantum-controlled form, namely, C U , but the time evolution op-erator is not. Adding a quantum control to a quantumgate– a task which we call controllization – is not trivialwhen U is unknown. In this paper we introduce univer-sal controllization , a quantum subroutine that approxi-mately implements controllization for unknown U .We introduce a d -dimensional ancillary system de-noted by H a = C d and define a unitary gate W U := C S ( I ⊗ U ⊗ I d ) C S on H c ⊗ H t ⊗ H a where C S is a uni-tary gate called the controlled-swap operation defined by C S | (cid:105)| ψ (cid:105)| φ (cid:105) = | (cid:105)| φ (cid:105)| ψ (cid:105) and C S | (cid:105)| ψ (cid:105)| φ (cid:105) = | (cid:105)| ψ (cid:105)| φ (cid:105) , forany | ψ (cid:105) , | φ (cid:105) ∈ C d , and I k denotes the k × k identity ma-trix. We call W U a classically conditioned quantum gate since it perfectly simulates C U when the control qubitis in a state | (cid:105) or | (cid:105) . But W U deviates from C U for ageneral input state | η (cid:105) = α | (cid:105) + β | (cid:105) in the control. Since W U | η (cid:105)| ψ (cid:105)| φ (cid:105) = α | (cid:105)| ψ (cid:105) U | φ (cid:105) + β | (cid:105) U | ψ (cid:105)| φ (cid:105) , the ancilla sys-tem is also entangled to the control and target systemsand thus decoherence occurs in the control-target systemin general. If we can prepare an eigenstate of U in theancilla system, exact implementations of C U is possible[12–14], but such implementations require knowledge on U . Other know controllization schemes [15, 16] also re-quire that the quantum gate is at least partially known.It is even proven that an exact controllization is impos-sible within quantum mechanics [14, 17]. These resultsare derived assuming that the input quantum gate is ablackbox. The unitary operator U ( t ), on the other hand,has a tunable parameter, namely, the evolution duration t . We exploit this feature and a decoupling method [18]used in quantum information theory to asymptoticallyimplement a universal controllization of U ( t ). The im-plementation accuracy of our controllization depends onthe maximum difference between any two eigenvalues of H .To reduce the decoherence by W U , we need to make theresulting state of the ancilla depend as little as possibleon the initial control-target state. Let us prepare theancilla in the completely mixed state I d /d , so that thestate of the ancilla remains the same at least when thecontrol qubit is in | (cid:105) or | (cid:105) for any given U . We considerthe reduced map on the control-target system,Γ U [ ρ ] := Tr H a (cid:104) W U ( ρ ⊗ I d /d ) W † U (cid:105) , (1)where ρ is a density matrix on C ⊗ H t . We call the mapΓ U as pseudo controllization . For ρ = | η (cid:105)(cid:104) η | ⊗ | ψ (cid:105)(cid:104) ψ | , wehaveΓ U [ | η (cid:105)(cid:104) η | ⊗ | ψ (cid:105)(cid:104) ψ | ] = C U ( | η (cid:105)(cid:104) η | ⊗ | ψ (cid:105)(cid:104) ψ | ) C † U + (cid:2) αβ ∗ | (cid:105)(cid:104) | ⊗ | ψ (cid:105)(cid:104) ψ | ( γ U − U † + c.c. (cid:3) , (2)where γ U = Tr [ U ] /d . The second term in Eq. (2) actsas a kind of phase damping noise on the control-targetsystem. The factor γ U − U from the ideal controllization. Wedefine the coherence factor a U := | γ U | and a phase factor e iϕ U := γ U / | γ U | . Notice that 1 − a U ≤ | γ U − | . Thus,the phase damping noise is minimized if we regard Γ U asan approximation of C U (cid:48) for U (cid:48) = e − iϕ U U . In a sense, Γ U implements a noisy controlled-unitary operation, wherethe magnitude of the noise is determined by a positivequantity 1 − a U .We further reduce the dependence of the ancilla on theinitial control-target state by use of a set { σ r } of unitaryoperations on the ancilla such that1 d (cid:88) r σ r W U ( ρ tot ⊗ I d /d ) W † U σ † r = Γ U [ ρ tot ] ⊗ I d /d. (3)Note that the ancilla is “refreshed” to the completelymixed state only by operations on the aniclla. (Such arandom operation has been extensively applied to ques-tions in quantum communication [18].) We divide W U ( t ) into m repetitions of W U ( t/m ) , each followed by the re-freshing operation (3). Here, m fixes the refresh rate.The strength of the noise after each refreshing opera-tion is O ( m ). Thus the total effect of the noise scales O ( m × m ) = O ( m ), which vanishes in the asymptoticlimit of m → ∞ . (see Appendix. , for details). Thisphenomenon is mathematically analogous to the quan-tum Zeno effect [19].We call this asymptotic implementation of acontrolled-unitary operation including the repeated re-freshing operation, universal controllization . For fi-nite m , the universal controllization approximates thecontrolled-unitary operation C U [ m ] ( t ) , where U [ m ] ( t ) = e − imϕ U ( t/m ) U ( t ). With m → ∞ , exp( imϕ U ( t/m ) ) con-verges to exp( − i Tr [ H ] t/d ). In a sense, universal con-trollization fixes the reference point of the energy of H so that Tr [ H ] = 0. A more detailed discussion of univer-sal controllization is presented in Appendix. . PME BY UNIVERSAL CONTROLLIZATION
A perfect PME for a system with a Hamiltonian H is distinguished from other quantum operations by twoproperties. First, the system remains in the same eigen-state when a PME is applied consecutively. Second, theoutcomes of the consecutive measurements are all pre-cisely equal to E k . The probability density p ( E | E k ) ofobtaining E as the outcome must be the delta function δ ( E ; E k ) := δ ( E − E k ). Conversely, the only measure-ment satisfying these properties is a perfect PME.A subtlety is that a perfect PME for H and for H − λ I should be considered equivalent, since two Hamiltonianswith different reference points of energy are physicallyequivalent. A measurement scheme is regarded as a per-fect PME for H if p ( E | E k ) = δ ( E ; E k − λ ) as long as λ is independent of k .Our PME protocol uses QPE on the time evolution op-erator U ( t ) with C U [ m ] ( t ) implemented by universal con-trollization. Here, the control qubits and ancilla of theuniversal controllization serve as the probe. The probe-target interaction is used to perform W U ( t/m ) , the re-freshing operations, and QFT. The lower figure in Fig. 1provides a conceptual diagram.In the ideal case of m → ∞ and N → ∞ , the modifiedQPE implements the projective measurement defined bythe spectral decomposition of ˜ U ( t ) = exp (cid:0) − i ˜ Ht (cid:1) , where˜ H := H − Tr [ H ] I . The outcome f gives − ˜ E k t (mod 2 π )for some energy eigenvalue ˜ E k of ˜ H .˜ E k cannot be uniquely determined from f for general t due to the periodicity of the phase function exp( iθ ). Letus restrict t so that ˜ E k ∈ ( π/t, − π/t ), namely,∆ max t ≤ π/ , (4)where ∆ max = max k,l (cid:12)(cid:12) ˜ E k − ˜ E l (cid:12)(cid:12) . The energy eigenvaluesare uniquely determined by E [ f ] = (cid:40) − πf /t f ∈ (cid:2) , (cid:1) − (2 πf − π ) /t f ∈ (cid:2) , (cid:1) . (5)Recall that the probability distribution of f is the deltafunction δ ( f − θ k / π ). Thus, p ( E | E k ) = δ ( E ; E k − Tr [ H ]),which is the desired function. The projection onto thecorresponding energy eigenspace is already guaranteedby QPE.For finite m and N , we continue to choose t accordingto Eq. (4) and estimate E k by Eq. (5) with f replaced by f ( n N ). The implemented measurement is an approxima-tion of a PME. A target initially in an energy eigenstate | E k (cid:105) results in the same state at the end of the scheme.One of the conditions for a perfect PME is still satisfied.Thus, the accuracy of the scheme is determined by howclose p ( E | E k ) for each | E k (cid:105) simulates a delta function δ ( E ; E k − λ ). IMPLEMENTATION ACCURACY AND TIMECOST
Recall that p N ( f ( n N ) | θ k ) in QPE needs to approachthe delta function δ ( f − θ k / π ) in N → ∞ to achieve theprojective measurement determined by U . If each C U inQPE is replaced by the adapted classical controllization(i.e., substituted by W U and an ancilla), p N ( f ( n N ) | θ k )does not converge to the delta function unless a U = 1.Let us denote by p [ m ] N ( f ( n N ) | E k ) the probability dis-tribution of f ( n N ) for a given m , N , and initialstate | E k (cid:105) . For a finite m , the universal control-lization approximately controllizes U (cid:48) = exp( − i ( Ht − mϕ U ( t/m ) I )). In this case, each run of the approximatedQPE provides an estimate for the eigenvalue correspond-ing to | E k (cid:105) , which is θ (cid:48) k = − E k t + mϕ U ( t/m ) (mod 2 π ).When N increases, the deviation of p [ m ] N ( f ( n N ) | E k )from p N ( f ( n N ) | θ (cid:48) k ) caused by the controllization er-ror prevents the function converging to a delta func-tion. (See Fig. 2.a) The deviation can be bounded by (cid:12)(cid:12)(cid:12) p [ m ] N ( f ( n N ) | E k ) − p N ( f ( n N ) | θ (cid:48) k ) (cid:12)(cid:12)(cid:12) ≤ (cid:15) for any (cid:15) > m is set to m ≥ (∆ max t ) N N − /(cid:15) (6)as shown in Appendix. .(See Fig. 2 b for examples.)For a given refresh rate m , each universal controlliza-tion makes m uses of W U ( t/m ) , where the total evo-lution duration ( t/m ) × m = t is independent of m .Hence, p [ m ] N ( f ( n N ) | E k ) can be brought arbitrarily closeto p N ( f ( n N ) | θ (cid:48) k ) without increasing the time cost. Thedistribution p N ( f ( n N ) | θ (cid:48) k ) is not a delta function for anyfinite N even with perfect controlled-unitary operations C U (i.e., infinite m ). The cost doubles for each controlqubit added, but the distance between p N ( f ( n N ) | θ (cid:48) k ) andthe delta function δ ( E ; ˜ E k ) is independent of the dimen-sion of the target. Hence, the implementation accuracyof PME can be improved without any dimension depen-dence. QUANTUM ALGORITHMIC ESTIMATION OFTHE ENERGY SCALE
We showed the existence of our PME protocol underthe assumption that ∆ max is known. The assumptioncan be relaxed to knowing an upper bound on ∆ max . Thebound may be estimated by quantum (process) tomog-raphy, but the tomography requires that a prior distri-bution of H is given. For a certain prior distribution,it is possible to estimate the bound by measuring the (a) p [ m ] N ( f ( n N ) | E k ) for a fixed m (b) p [ m ] N ( f ( n N ) | E k ) for an adaptively chosen m FIG. 2. (Color online) Plots of probability distributions p [ m ] N ( f ( n N ) | E k ) and their envelope functions for target Hamil-tonian H = − (cid:80) λ =0 | E ( λ )0 (cid:105)(cid:104) E ( λ )0 | + | E (cid:105)(cid:104) E | , t = 0 . · π , andsetting E k = E . Each marker represents p [ m ] N ( f ( n N ) | E k ) offinding each outcome by a single round of the PME scheme.Fig.2 (a) presents p [ m ] N ( f ( n N ) | E k ) for N = 2 , , m = 8. Fig.2 (b) presents p [ m ] N ( f ( n N ) | E k ) for N = 2 , , m is adoptively chosen as asmallest integer satisfying m ≥ (∆ max t ) N N − /(cid:15) and targeterror (cid:15) is set to 0 .
25. In all cases, markers corresponding toprobability less than one tenth of the target error (0 . coherence factor a U . We observe that a U approaches 1as the product ∆ max t decreases to 0. Thus, when a U isestimated to be close to 1, it is possible that ∆ max t issufficiently small. While this is not true for some Hamil-tonians, the probability of such “error” decreases expo-nentially in the dimension d of the target for a particularclass of prior distribution (see Appendix. ). Hence, wecan reliably estimate a U .To estimate a U , we modify the quantum algorithm pre-sented in Ref. [11]. The original algorithm outputs thetrace Tr [ U ] of an input unitary U , provided that thecorresponding C U is available. In our problem, we re-place C U with W U . With this modification, the originalalgorithm returns | Tr [ U ] | (See Appendix. , for details),thus we obtain a U since a U = | Tr [ U ] | /d . Clearly, thismodified algorithm estimates a U much more efficientlythan process tomography. Conclusion.—
In this letter, we presented an implemen-tation protocol for a projective measurement of energyon a system driven by an unknown Hamiltonian with agiven energy scale. The implementation time cost of theprotocol is independent of the dimension of the systemunlike the one based on quantum process tomography.The protocol is based on a computational quantum al-gorithm called quantum phase estimation (QPE). We in-troduced universal controllization to make the computa-tional algorithm executable without stopping the evolu-tion of the target system. Another computational quan-tum algorithm is shown to be effective in estimating theenergy scale with a suitable modification. This motivatesthe search for further applications of quantum algorithmsoutside their original computational context.
Acknowledgments:
The authors thank T. Sugiyamaand H. Nishimura for their insights and expertise. Thiswork is supported by the Project for Developing Innova-tion Systems of MEXT, Japan, the Global COE Pro-gram of MEXT Japan, and JSPS KAKENHI (GrantNo. 23540463, No. 23240001, and No. 26330006). Theauthors also gratefully acknowledge the ELC project(Grant-in-Aid for Scientific Research on Innovative AreasMEXT KAKENHI (Grant No. 24106009)) for encourag-ing the research presented in this paper. After the com-pletion of our work, we were notified that our algorithmfor calculating | Tr [ U ( t )] | based on DQC1 has been inde-pendently discovered by J. Thompson, M. Gu, K. Modi,and V. Vedral in “Quantum Computing with Black-boxSubroutines” [20]. We thank these authors for drawingtheir work to our attention. Elements of quantum phase estimation
This section provides details of quantum phase estima-tion (QPE) described in the second and the third sectionsin the main article on
Projective measurement by QPE and
QPE and universal controllization . QPE plays a cru-cial role in our protocol for projective measurement ofenergy (PME). A quantum circuit representation of thealgorithm of QPE, the probability distribution of out-comes and the transformed state corresponding to eachoutcome by QPE are presented in Sec. . In the main pa-per, we referred that the probability distribution of theoutcome converges to a delta function at the limit wherethe number of control qubits goes infinity. In Sec. , wegive the mathematical formulation of the statement.
Probability distribution of outcomes and statechange induced by QPE
QPE is designed so that each run returns a good esti-mate for one of the eigenvalues of a given unitary oper-ator U . QPE (originally proposed in [10]) is usually de-scribed in the state-vector formalism. In this subsection,we provide another description based on the density-matrix formalism to facilitate the comparison with the approximate QPE using universal controllization pre-sented in Appendix. .The circuit representation of QPE is given in Fig. 3.Consider a target system C d and a control system con-sisting of N -qubit systems C N . We set a basis of a qubitsystem C and denote the basis by {| (cid:105) , | (cid:105)} .First, we initialize the control and target system as | . . . (cid:105)(cid:104) . . . | ⊗ | θ k (cid:105)(cid:104) θ k | , (7)on C N ⊗ C d , where | . . . (cid:105) := | (cid:105) ⊗ · · · ⊗ | (cid:105) and | θ k (cid:105) is an eigenvector corresponding to eigenvalue e iθ k of U .The Hadamard gate H is then applied to each controlqubit. Note that H achievesH | (cid:105) = ( | (cid:105) + | (cid:105) ) / √ , (8)H | (cid:105) = ( | (cid:105) − | (cid:105) ) / √ . (9)The state after this operation is given by12 N (cid:88) a ,a ,...a N b ,b ,...,b N | a a . . . a N (cid:105)(cid:104) b b . . . b N | ⊗ | θ k (cid:105)(cid:104) θ k | , (10)where a l , b l ∈ { , } .A controlled-unitary operation C U of an unitary oper-ation U is defined as C U := | (cid:105)(cid:104) | ⊗ I d + | (cid:105)(cid:104) | ⊗ U (11)on C ⊗ C d . Here, I d denotes the d × d identity matrix.The superoperator representation C U , corresponding to C U , is defined as C U [ ρ ] := C U ρC † U . (12)We choose the l -th control qubit and the target systemand apply the controlled-unitary operation C U l − for all1 ≤ l ≤ N . This transforms the state to12 N (cid:88) a ,a ,...a N b ,b ,...,b N N (cid:89) l =1 exp (cid:0) i l − ( a l − b l ) θ k (cid:1) × | a a . . . a N (cid:105)(cid:104) b b . . . b N | ⊗ | θ k (cid:105)(cid:104) θ k | . (13)Finally, the quantum Fourier transformation is appliedand then the control qubits are measured in the compu-tational basis (cid:40) | n N (cid:105) = | n n . . . n N (cid:105) (cid:12)(cid:12)(cid:12) n N = N (cid:88) l =1 n l · l − (cid:41) . (14)The Fourier transformation and the measurement in thecomputational basis together are equivalent to perform-ing a projective measurement on the state (13) in theFourier basis, i.e., (cid:8) | f ( n N ) (cid:105)| ≤ n N < N (cid:9) , (15) control 1 control 2 control 3 target | θ k >|0>|0>|0> U U U HHH
QFT
FIG. 3. A quantum circuit representation of QPE. The boxQFT denotes the quantum Fourier transformation. The finalmeasurement is performed in the computational basis. where f ( n N ) := n N / N and | f ( n N ) (cid:105) := N − (cid:88) n (cid:48) N =0 exp ( i πf ( n N ) n (cid:48) N ) √ N | n (cid:48) N (cid:105) . (16)The probability distribution p N ( f ( n N ) | θ k ) of obtainingthe state | f ( n N ) (cid:105) is calculated as p N ( f ( n N ) | θ k )= 12 N N (cid:89) l =1 (cid:88) a,b ∈{ , } exp (cid:2) i l − ( a − b )( θ k − πf ( n N )) (cid:3) = 12 N N (cid:89) l =1 (cid:2) (cid:2) l − ( θ k − πf ( n N )) (cid:3)(cid:3) . (17)This simplifies to p N ( f ( n N ) | θ k ) = (cid:32) sin (cid:2) N ( θ k − πf ( n N )) / (cid:3) N sin (cid:2) ( θ k − πf ( n N )) / (cid:3) (cid:33) , (18)using 1 + cos(2 l − x ) = 12 (cid:18) sin(2 l − x )sin(2 l − x ) (cid:19) (19)to Eq. (17).If we apply QPE to an arbitrarily superposed inputstate | φ (cid:105) = (cid:80) k α k | θ k (cid:105) , where (cid:80) k | α k | = 1, the probabil-ity distribution p N ( f ( n N ) | φ ) of obtaining the outcomes { n , · · · , n N } represented in terms of f ( n N ) is given by p N ( f ( n N ) | φ ) = (cid:88) k | α k | p N ( f ( n N ) | θ k ) . (20)For the given outcome f ( n N ), the corresponding outputstate of the target system can be calculated as | φ (cid:48) f ( n N ) (cid:105) = (cid:88) k α k (cid:115) p N ( f ( n N ) | θ k ) p N ( f ( n N ) | φ ) e iG ( θ i ,f ( n N )) | θ k (cid:105) , (21) where G ( θ k , f ( n N )) = (cid:0) N − (cid:1) ( θ k − πf ( n N )) / . Thus if the distribution p N ( f ( n N ) | θ k ) converges to thedelta function δ ( f − θ k / π ) for N → ∞ , then the outputstate | φ (cid:48) f ( n N ) (cid:105) converges to a particular eigenstate | θ k (cid:105) .In other words, in the limit of N → ∞ , we only obtain f = θ k / π with the target system in the correspondingeigenstate. We see that QPE implements a projectivemeasurement in the eigenbasis of U for N → ∞ . If theunitary U is generated by a Hamiltonian H as U ( t ) =exp ( − iHt ), QPE implements projective measurement ofenergy (PME) of H up to ambiguity due to the phaseperiodicity. Convergence of p N ( f ( n N ) | θ k ) to a delta function We assumed that p N ( f ( n N ) | θ k ) converges to the deltafunction δ ( f − θ k / π ) for N → ∞ in the last subsection.For each θ k and any finite N , f ( n N ) is a discrete randomvariable over (cid:8) n N / N | n N = 0 , . . . , N − (cid:9) , distributedaccording to p N ( f ( n N ) | θ k ). In contrast, f is a continuous random variable over real numbers x in 0 ≤ x ≤
1. Inthe followings, we introduce a precise statement of theconvergence to justify the assumption. The convergenceof a discrete random variable to a continuous one can beformulated with distribution functions [21].Let [ a, b ] denote the set of real numbers x such that a ≤ x ≤ b . For any A ⊂ [0 , µ N ( A ) by µ N ( A ) = (cid:88) { n N | f ( n N ) ∈ A } p N ( f ( n N ) | θ k ) . (22)If A = [ a, b ] for 0 ≤ a ≤ b < θ k / π , we can bound µ N ( A )as µ N ( A ) ≤ N A (cid:18) N sin[( θ k − πb ) / (cid:19) , (23)where N A is the number of f ( n N ) satisfying f ( n N ) ∈ A .Since N A ≤ N ( b − a ) + 1 , we have µ N ( A ) ≤ [( θ k − πb ) /
2] 2 N ( b − a ) + 12 N . (24)Similarly for A = [ a, b ] and θ k / π < a ≤ b ≤
1, we obtain µ N ( A ) ≤ [( θ k − πa ) /
2] 2 N ( b − a ) + 12 N . (25)Let F N ( f max ) be the distribution function of f ( n N ) fora given N , i.e., F N ( f max ) = P ( f ( n N ) ≤ f max ) (26)= (cid:88) f ( n N ) ∈ [0 ,f max ] p N ( f ( n N ) | θ k ) (27) (a) controlancillasystem U(t/m)
I/d = (b) controlancillasystem r ×m I/d W U(t/m)
FIG. 4. (a) A quantum circuit representation of the classicallyconditioned quantum gate W U ( t/m ) . (b) A quantum circuitrepresentation of the algorithm implementing the universalcontrollization of U ( t ). The gate σ r is chosen uniformly ran-domly for each iteration from the set S defined in Lemma 1.The controlled-swap operation and the random unitary oper-ations are to be performed instantaneously. and F ( f max ) be that of the continuous variable f , givenby F ( f max ) = P ( f ( n N ) ≤ f max ) (28)= (cid:90) f max δ ( θ k − πf ) df. (29)For 0 ≤ f max ≤ θ k / π , we have from Ineq. (24) that F N ( f max ) ≤ [( θ k − πf max ) /
2] 2 N f max + 12 N , (30)therefore, lim N →∞ F N ( f max ) = F ( f max ) = 0 . (31)For θ k / π < f max ≤
1, we see from Ineq. (25) that F N ( f max ) ≥ − [( θ k − πf max ) /
2] 2 N (1 − f max )+12 N , (32)which implieslim N →∞ F N ( f max ) = F ( f max ) = 1 . (33)Therefore, for all points at which F ( f ) is continuous, F N ( f ) converges to F ( f ), thus the random variable f ( n N ) converges to f in distribution for N → ∞ . Universal controllization
This section is related to the fourth section of the maintext on
QPE and universal controllization’ . In Sec. , weprovide a mathematical statement which supports theexistence of the refresh operations used in universal con-trollization. We derive a description of the superoperator(CPTP map) implemented by the universal controlliza-tion in Sec. , and analyze the error in controllization fora given refresh rate m in Sec. . We also show how to ob-tain the limit lim m →∞ mϕ U ( t/m ) in Sec. , which appearsin universal controllization. We denote the k × k identitymatrix by I k . Preliminary
We present mathematical relations that justify the re-freshing operation (3) in the main text.
Lemma 1.
Let H be a d -dimensional Hilbert space and G a finite group with a d × d unitary irreducible repre-sentation Σ . If a set S := { σ , σ , . . . , σ D } of unitarieson H satisfies Σ( g ) (cid:18) D (cid:88) r =1 σ r Aσ † r (cid:19) Σ( g ) † = D (cid:88) r =1 σ r Aσ † r (34) for any operator A on H and g ∈ G , then D (cid:88) r σ r Aσ † r = Tr [ A ] d · I d (35) for any given operator A . Let us define an operator ˜ A := (cid:80) Dr =1 σ r Aσ † r . Equa-tion (34) implies thatΣ( g ) ˜ A = ˜ A Σ( g ) . (36)By Schur’s lemma, such an operator ˜ A satisfies˜ A = a · I d . (37)Since Tr (cid:2) σ r Aσ † r (cid:3) = Tr [ A ], we have a = Tr (cid:104) ˜ A (cid:105) d = 1 d · D (cid:88) r Tr (cid:2) σ r Aσ † r (cid:3) = Tr [ A ] d , (38)which proves Lemma 1.We also introduce the following corollary of thislemma. Corollary 1.
Let H be as defined in Lemma 1 and H (cid:48) bea d (cid:48) -dimensional Hilbert space. For any operator M on H (cid:48) ⊗ H we have that D D (cid:88) r =1 ( I d (cid:48) ⊗ σ r ) M ( I d (cid:48) ⊗ σ r ) † = Tr H [ M ] ⊗ I d d , (39) where σ r is taken from S as defined in Lemma 1. The proof follows almost immediately from Lemma 1,since any operator M on H (cid:48) ⊗ H can be decomposed as M = (cid:88) k A (cid:48) k ⊗ A k , (40)where A (cid:48) k and A k are operators on H (cid:48) and H , respectively. Superoperator description
We are now ready to derive the superoperator imple-mented by the universal controllization. Consider a uni-tary operation U ( t ) := exp( − iHt ) generated by a Hamil-tonian H on H t = C d . A quantum circuit representationof the algorithm is presented in Fig. 4(b). It uses onecontrol qubit and a d -dimensional ancilla. The respec-tive Hilbert spaces are denoted by H c and H a .Let ρ = (cid:80) k,j =0 , | k (cid:105)(cid:104) j | ⊗ ρ kj on H c ⊗ H t be the initialstate of the control-target system. The initial state of thetotal system including the ancilla is given by the densitymatrix ρ tot = ρ ⊗ I d d (41)on H c ⊗ H t ⊗ H a . The algorithm first performs the clas-sical conditioned quantum gate W U ( t/m ) = C S · ( I ⊗ U ( t/m ) ⊗ I d ) · C S . (42)followed by the refreshing operation σ r on the ancilla.Here, C S is the controlled-swap operation, defined in themain text. Figure 4(b) describes the circuit for pseudocontrollization of U ( t/m ). Note that we take W U ( t/m ) as a unitary on H c ⊗ H t ⊗ H a , while the subsystems inthe figure are arranged in the order of the control, an-cilla, and target, which in the figure is labeled “system”.These two quantum operations are repeated in the sameorder for m times. For each iteration, σ r is chosen uni-formly randomly from S defined in Lemma 1. We seefrom Corollary 1 that the first iteration yields1 D (cid:88) r ( I ⊗ I d ⊗ σ r ) W U ( tm ) ρ tot W † U ( tm ) ( I ⊗ I d ⊗ σ r ) † = Tr H a (cid:20) W U ( tm ) ρ tot W † U ( tm ) (cid:21) ⊗ I d d = Γ U ( tm )[ ρ ] ⊗ I d d , (43)where the summation over r is to reflect that σ r is chosenuniformly randomly. Thus, m iterations achieve ρ ⊗ I d d → Γ mU ( tm )[ ρ ] ⊗ I d d . (44)Simple algebra will show that W U ( tm ) ρ tot W † U ( tm )= 1 d (cid:88) k,j | k (cid:105)(cid:104) j | ⊗ U (cid:18) ktm (cid:19) ρ kj U † (cid:18) jtm (cid:19) ⊗ U (cid:18) ( j − k ) tm (cid:19) . (45)Therefore, the first iteration can be seen as transforma- tion ρ → ρ , (46) ρ → ρ (cid:16) γ U ( t/m ) U † ( t/m ) (cid:17) , (47) ρ → (cid:16) γ ∗ U ( t/m ) U ( t/m ) (cid:17) ρ , (48) ρ → U ( t/m ) ρ U † ( t/m ) , (49)where γ U ( τ ) := Tr [ U ( τ )] /d. (50)We have thusΓ mU ( tm )[ ρ ] = C U ( t ) ρC † U ( t ) + (cid:104) | (cid:105)(cid:104) | ⊗ ρ (cid:16) γ mU ( t/m ) − (cid:17) U † + c.c. (cid:105) , (51)which is the superoperator implemented by the universalcontrollization. Accuracy of the universal controllization
The previous subsection shows that the universal con-trollization is a map from a quantum gate U ( t ) to thesuperoperator Γ mU ( t/m ) . The ideal universal controlliza-tion would be a map from U ( t ) to the superoperator C U ( t ) [ ρ ] := C U ( t ) ρC † U ( t ) . (52)Let us evaluate the accuracy of the universal controlliza-tion for a given m as a distance between the maps Γ mU ( t/m ) and C U ( t ) . Theorem 1.
For any m ∈ N and a unitary operator U ( t ) = exp( − iHt ) generated by a Hermitian operator H on C d and t ∈ R , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ mU ( t/m ) − C U [ m ] ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:5) = 1 − a mU ( t/m ) , (53) where a U ( τ ) := (cid:12)(cid:12)(cid:12)(cid:12) Tr [ U ( τ )] d (cid:12)(cid:12)(cid:12)(cid:12) , U [ m ] ( t ) := e − imϕ U ( t/m ) U ( t ) , (54) for ϕ U ( τ ) defined by e iϕ U ( τ ) := γ U ( τ ) a U ( τ ) . (55)The diamond norm ||·|| (cid:5) [22] in this theorem is a normfor superoperator, which takes into account when the su-peroperators is extendend to act on a part of a largerHilbert space than for which it is originally defined. It isoften used to evaluate the difference between two CPTPmaps in the context of quantum information.A superoperator S on a Hilbert space H acting on anextended system H ⊗ H (cid:48) satisfies ||S|| op < || ( S ⊗ id H (cid:48) ) || op , (56)where the operator norm ||S|| op is the maximum of thetrace norm of S [ A ] for an operator A under the condition || A || tr = 1 and id H (cid:48) denotes the identity superoperatoron H (cid:48) . The trace norm is defined as || A || tr = Tr (cid:2) AA † (cid:3) .Since ||S ⊗ id H (cid:48) || op ≤ ||S ⊗ id H || op holds for any Hilbertspace H (cid:48) , it is enough to consider ||S ⊗ id H || op to bound ||S ⊗ id H (cid:48) || for any H (cid:48) . The diamond norm ||·|| (cid:5) of asuperoperator S on the Hilbert space H is defined as ||S|| (cid:5) := ||S ⊗ id H || op . (57)The following lemma is convenient for calculating thediamond norm. Lemma 2.
Any Hermitian preserving superoperator Λ on the Hilbert space H satisfies || Λ || (cid:5) = max P ∈P || (Λ ⊗ id H ) P || tr , (58) where P is a set of rank-1 projectors on H ⊗ H . See Ref. [22] for a proof.Let us prove Theorem 1. To calculate the diamondnorm, we search for rank-1 projectors on ( H c ⊗ H t ) ⊗ :=( H c ⊗ H t ) ⊗ ( H c ⊗ H t ) that gives the largest trace normafter Γ mU ( t/m ) − C U [ m ] ( t ) is applied. Any rank-1 projectoron ( H c ⊗ H t ) ⊗ is given by | Ψ (cid:105)(cid:104) Ψ | for some vector | Ψ (cid:105) in( H c ⊗ H t ) ⊗ .All vectors in ( H c ⊗ H t ) ⊗ can be represented as | Ψ (cid:105) = α | (cid:105)| ψ (cid:105) + β | (cid:105)| φ (cid:105) , where {| (cid:105) , | (cid:105)} is the computationalbasis of the first control qubit system H c , | ψ (cid:105) and | φ (cid:105) arenormalized vectors in H t ⊗H c ⊗H t , and α, β satisfy | α | + | β | = 1. As a block matrix, the projector is representedby | Ψ (cid:105)(cid:104) Ψ | = (cid:18) | α | | ψ (cid:105)(cid:104) ψ | αβ ∗ | ψ (cid:105)(cid:104) φ | α ∗ β | φ (cid:105)(cid:104) ψ | | β | | φ (cid:105)(cid:104) φ | (cid:19) . (59)The upper left block corresponds to the | (cid:105)(cid:104) | elementof the first system. The upper right block is the | (cid:105)(cid:104) | element, and the other blocks are defined similarly. Theprojector | Ψ (cid:105)(cid:104) Ψ | is transformed by the maps C U [ m ] ( t ) andΓ mU ( t/m ) as (cid:0) C U [ m ] ( t ) ⊗ id C ⊗ C d (cid:1) [ | Ψ (cid:105)(cid:104) Ψ | ]= (cid:18) | α | | ψ (cid:105)(cid:104) ψ | αβ ∗ | ψ (cid:105)(cid:104) φ | U [ m ] ( t ) † α ∗ βU [ m ] ( t ) | φ (cid:105)(cid:104) ψ | | β | U [ m ] ( t ) | φ (cid:105)(cid:104) φ | U [ m ] † ( t ) (cid:19) (60)and (cid:16) Γ mU ( tm ) ⊗ id C ⊗ C d (cid:17) [ | Ψ (cid:105)(cid:104) Ψ | ]= | α | | ψ (cid:105)(cid:104) ψ | γ mU ( tm ) αβ ∗ | ψ (cid:105)(cid:104) φ | U ( t ) † γ ∗ mU ( tm ) α ∗ βU ( t ) | φ (cid:105)(cid:104) ψ | | β | U ( t ) | φ (cid:105)(cid:104) φ | U ( t ) † . (61) Note that γ mU ( tm ) = exp (cid:16) imϕ U ( tm ) (cid:17) a mU ( tm ) , (62) U [ m ] ( t ) = exp (cid:16) − imϕ U ( tm ) (cid:17) U ( t ) . (63)A direct calculation will show that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) αβ ∗ | ψ (cid:105)(cid:104) φ | U [ m ]( t ) † α ∗ βU [ m ] ( t ) | φ (cid:105)(cid:104) ψ | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) tr = 2 | αβ | . (64)Therefore, the norm of interest is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C U [ m ] ( t ) − Γ mU ( tm ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:5) = 2 (cid:16) − a mU ( tm ) (cid:17) max α,β | αβ | = 1 − a mU ( tm ) , (65)where we have used the normalization condition for α and β to obtain the last equality. This proves Theorem 1.The distance between C U [ m ] ( t ) and Γ mU ( t/m ) approaches0 as m increases. Here is an intuitive argument: First, a U ( t/m ) = (cid:114) d Tr [ U ( t/m )] · Tr [ U ( t/m )] ∗ = (cid:115) d (cid:88) k e − iE k t/m · (cid:88) l e iE l t/m = (cid:115) d (cid:88) k,l e i ( E k − E l ) t/m = (cid:118)(cid:117)(cid:117)(cid:116) d (cid:32)(cid:88) k>l cos (cid:2) ( E k − E l ) t/m (cid:3)(cid:33) + 1 d . (66)We invoke the Taylor expansion of cos( αx ) to the secondorder in x . cos( ax ) = 1 − α x + O ( x ) , (67)and let α kl = ( E k − E l ) t and x = 1 /m . Under thesenotations, a U ( t/m ) = (cid:118)(cid:117)(cid:117)(cid:116) − d (cid:32)(cid:88) k>l α kl (cid:18) m (cid:19) (cid:33) + O ( m − )= 1 − C (cid:18) m (cid:19) + O ( m − ) , (68)where the last equality is derived using the Taylor ex-pansion of √ x to the second order of x with C = (cid:0)(cid:80) k>l α kl (cid:1) / d . Finally, we Taylor expand (1 + x ) m tothe second order in x and set x = C/m to obtain a mU ( t/m ) = 1 − m · C (cid:18) m (cid:19) + O ( m − ) , (69)which converges to 1 as m → ∞ . Thus the distance(65) converges to 0. This property is mathematicallyanalogous to the quantum Zeno effect [19].0 Convergence of the phase factor
We stated at the end of the fourth section in the maintext (“Universal controllization”) that the phase shift mϕ U ( t/m ) induced by the universal controllization sat-isfies lim m →∞ e imϕ U ( t/m ) = e − i Tr[ H ] t/d . (70)A proof is as follows. Since a U ( t/m ) = 1 + O (1 /m ), thecoherence factor can be sorted by the order of m as (cid:16) e iϕ ( t/m ) (cid:17) m = (cid:18) − i Tr [ H ] d tm + O (cid:18) m (cid:19)(cid:19) m . (71)Hence, we can conclude that e imϕ ( t/m ) = (cid:18) − i Tr [ H ] d tm (cid:19) m + O (cid:18) m (cid:19) (72)= e − i Tr[ H ] t/d + O (cid:18) m (cid:19) . (73) Algorithm for directly evaluating the accuracy ofcontrollization
In the main article, universal controllization is intro-duced as a subroutine for PME. However, applicationsof universal controllization is not limited to PME, sincemany algorithms and protocols in quantum informationutilize controlled-unitary operations. In this section,we introduce a quantum algorithm that directly evalu-ates the accuracy of universal controllization given byEq. (65).Figure 5 gives a quantum circuit representation of thealgorithm. The total system consists of three subsystems,namely, the control ( H c ), target ( H t ), and ancilla ( H a ),with dimension 2, d , and d , respectively. We shall use I k to denote the k × k identity matrix. The system isprepared in the state | (cid:105)(cid:104) | ⊗ I d d ⊗ I d d (74)on H c ⊗ H t ⊗ H a . We denote the Pauli X, Y, and Zmatrix as σ x , σ y , and σ z , respectively, whose matrix rep-resentation is σ x = | (cid:105)(cid:104) | + | (cid:105)(cid:104) | (75) σ y = − i | (cid:105)(cid:104) | + i | (cid:105)(cid:104) | (76) σ z = | (cid:105)(cid:104) | − | (cid:105)(cid:104) | . (77)Our goal is to convert the state of the control qubit to ρ m = 12 (cid:16) I + a mU ( t/m ) σ z (cid:17) (78)and obtain the expectation value for σ z , i.e.,Tr[ ρ m σ z ] = a mU ( t/m ) . (79) controlancillasystem I/dI/d ×m W U(t/m) rs H H {0,1} |0 〉 FIG. 5. A quantum circuit representation of the algorithmthat estimates the controllization error of the universal con-trollization due to finite refresh rate m . First, we apply the Hadamard gate H on the controlqubit. This transforms the total state to I + σ x ⊗ I d d ⊗ I d d . (80)Consider a set of unitary operations S = { σ r | ≤ r ≤ D } on C d as defined in Lemma 1 and randomly choose aunitary operation σ rs := σ r ⊗ σ s . (81)We perform the unitary operation W U ( t/m ) on the stateEq. (74), which is followed by I d ⊗ σ rs . Each of theseoperations is applied m times in total, while σ rs is chosenat random for each repetition.Note that W U ( σ x ⊗ I d ⊗ I d ) W † U = σ x ⊗ (cid:0) U ⊗ U † + U † ⊗ U (cid:1) − iσ y ⊗ (cid:0) U ⊗ U † − U † ⊗ U (cid:1) . (82)It is also easy to see that1 D (cid:88) r,s ( I ⊗ σ rs ) W U ( σ x ⊗ I d ⊗ I d ) W † U ( I ⊗ σ rs ) † = | Tr [ U ] | d · σ x ⊗ I d ⊗ I d = a U · σ x ⊗ I d ⊗ I d . (83)The first iteration of W U ( t/m ) and I ⊗ σ rs converts thestate Eq. (74) to I + a U ( t/m ) σ x ⊗ I d d ⊗ I d d . (84)The next iteration changes a U ( t/m ) σ x to a U ( t/m ) σ x .Therefore, m iterations create the state I + a mU ( t/m ) σ x ⊗ I d d ⊗ I d d . (85)1We have the desired state (78) with another Hadamardgate on the control qubit.The expectation value (79) is calculated from the equa-tion Tr[ ρ m σ z ] = (cid:104) | ρ m | (cid:105) − (cid:104) | ρ m | (cid:105) . (86)Each term in the right hand side is the probability ofobtaining the outcome 0 and 1, respectively, from themeasurement on the control qubit in the computationalbasis. Thus, our algorithm calculates the distance (65). Approximated QPE with universal controllization
This section provides a supplemental material for thefifth section in the main article on
Implementation ac-curacy and time cost . In this section, we follow the cal-culations evaluating the error of approximated QPE im-plemented by universal controllization. The error is de-fined as the deviation of the probability distribution ofQPE from the ideal case. In Sec. , we first derive theprobability distribution of the approximated QPE. Thenwe evaluate the deviation of the probability distributionfrom the ideal distribution in Sec. .
Probability of outcomes
Let us calculate the probability distribution p [ m ] N ( f ( n N ) | E k ) of QPE with universal controlliza-tion. The initial state of the control ( H cont ) and targetsystem is given by | . . . (cid:105)(cid:104) . . . | ⊗ | E k (cid:105)(cid:104) E k | , (87)on H cont ⊗ H t . We note that dim H cont = 2 N .At the first step of the algorithm, the Hadamard gateis applied to each control qubit system. The state afterthis operation is given by12 N (cid:88) a ,a ,...a N b ,b ,...,b N | a a . . . a N (cid:105)(cid:104) b b . . . b N | ⊗ | E k (cid:105)(cid:104) E k | (88)where a l , b l ∈ { , } .At the second step, the universal controllization mapΓ mU l − ( t/m ) is applied on the pair of the l -th control qubitand the target system, for all 1 ≤ l ≤ N . These opera-tions transform the state to12 N (cid:88) a ,...a N b ,...,b N N (cid:89) l =1 a m l − | a l − b l | U ( tm ) × exp (cid:16) i l − ( a l − b l ) (cid:16) − E k t + mϕ U ( tm ) (cid:17)(cid:17) × | a , a . . . a N (cid:105)(cid:104) b b . . . b N | ⊗ | E k (cid:105)(cid:104) E k | . (89) Using the periodicity of the phase function and θ (cid:48) k = − E k t + mϕ U ( tm ) (mod 2 π ), the above equation simplifiesto 12 N (cid:88) a ,...a N b ,...,b N N (cid:89) l =1 a m l − | a l − b l | U ( tm ) exp (cid:0) i l − ( a l − b l ) θ (cid:48) k (cid:1) × | a , a . . . a N (cid:105)(cid:104) b b . . . b N | ⊗ | E k (cid:105)(cid:104) E k | . (90)Unlike Eq. (13), the coherence factor a U ( tm ) appears inEq. (90).In the final step, the inverse quantum Fourier transfor-mation is applied and the control qubits are measured inthe computational basis. This is equivalent to perform-ing a projective measurement in the Fourier basis (16).The probability p [ m ] N ( f ( n N ) | E k ) of obtaining f ( n N ) bythe measurement (cid:8) | f ( n N ) (cid:105)| ≤ n N < N (cid:9) on a densityoperator ρ is given by (cid:104) f ( n N ) | ρ | f ( n N ) (cid:105) . Therefore, ac-cording to Eq. (90) the probabilty distribution of QPEwith universal controllization satisfies p [ m ] N ( f ( n N ) | E k )= 12 N (cid:88) a ,...a N b ,...,b N N (cid:89) l =1 a m l − | a l − b l | U ( tm ) (91) × exp (cid:0) i l − ( a l − b l ) ( θ (cid:48) k − πf ( n N )) (cid:1) = 12 N N (cid:89) l =1 (cid:88) a,b =0 , a m l − | a − b | U ( tm ) × exp (cid:0) i l − ( a − b ) ( θ (cid:48) k − πf ( n N )) (cid:1) (92)= 12 N N (cid:89) l =1 (cid:104) a m l − U ( tm ) cos (cid:2) l − ( θ (cid:48) k − πf ( n N )) (cid:3)(cid:105) . (93) Accuracy of the approximated QPE
The error of approximated QPE is evaluated as (cid:12)(cid:12)(cid:12) p [ m ] N ( f ( n N ) | E k ) − p N ( f ( n N ) | θ (cid:48) k ) (cid:12)(cid:12)(cid:12) . To calculate this er-ror, we define the following quantity δ l representing theeffect of the phase damping noise, δ l = 1 − a m l − U ( t/m ) . (94)The probability distribution p [ m ] ( f ( n N ) | E k ) can be ex-pressed as p [ m ] N ( f ( n N ) | E k ) = N (cid:89) l =1 ( µ l − δ l ν l ) , (95)where µ l = 1 + cos 2 l − ( θ (cid:48) k − πf ( n N ))2 (96) ν l = cos 2 l − ( θ (cid:48) k − πf ( n N ))2 . (97)2We define a set K ( l ) of subsequences of { , , . . . , N } of length l , such that K ( l ) := (cid:110) { m , m , . . . , m l } (cid:12)(cid:12) ∀ i, ≤ m i < m i +1 ≤ N (cid:111) . (98)The decomposition of Eq. (95) is given as p [ m ] N ( f ( n N ) | E k )= N (cid:89) l =1 µ l + N (cid:88) l =1 (cid:88) K∈ K ( l ) (cid:89) i/ ∈K µ i (cid:89) j ∈K δ j ν j . (99)Note that the first term is p N ( f ( n N ) | θ (cid:48) k ). Then (cid:12)(cid:12)(cid:12) p [ m ] N ( f ( n N ) | E k ) − p N ( f ( n N ) | θ (cid:48) k ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) l =1 (cid:88) K∈ K ( l ) (cid:89) i/ ∈K µ i (cid:89) j ∈K δ j ν j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (100)By the triangular inequality, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) l =1 (cid:88) K∈ K ( l ) (cid:89) i/ ∈K µ i (cid:89) j ∈K δ j ν j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N (cid:88) l =1 (cid:88) K∈ K ( l ) (cid:89) i/ ∈K | µ i | (cid:89) j ∈K δ j | ν j | . (101)Since | µ l | ≤ | ν l | ≤ /
2, we arrive at (cid:12)(cid:12)(cid:12) p [ m ] N ( f ( n N ) | E k ) − p N ( f ( n N ) | θ (cid:48) k ) (cid:12)(cid:12)(cid:12) ≤ N (cid:88) l =1 (cid:88) K∈ K ( l ) (cid:81) j ∈K δ j l . (102)Let us assume the condition δ N = 1 − a m N − U ( t/m ) ≤ δ, (103)is satisfied for a fixed, then δ l ≤ δ is satisfied for 1 ≤ l ≤ N since a mU ( t/m ) ≤
1. Therefore (cid:12)(cid:12)(cid:12) p [ m ] N ( f ( n N ) | E k ) − p N ( f ( n N ) | θ (cid:48) k ) (cid:12)(cid:12)(cid:12) ≤ N (cid:88) l =1 (cid:88) K∈ K ( l ) δ l l = N (cid:88) l =1 N ! l !( N − l )! · δ l l = (cid:18) δ (cid:19) N − . (104)Here we introduce the following lemma. Lemma 3.
For any ≤ a ≤ and ≤ r ≤ r (cid:48) , (cid:16) − ar (cid:17) r ≤ (cid:16) − ar (cid:48) (cid:17) r (cid:48) , (105) (cid:16) ar (cid:17) r ≥ (cid:16) ar (cid:48) (cid:17) r (cid:48) . (106) This lemma can be easily checked by the following dif-ferential relation, ddr (cid:16) − ar (cid:17) r = ar (cid:16) − ar (cid:17) r − ≥ , (107) ddr (cid:16) ar (cid:17) r = − ar (cid:16) ar (cid:17) r − ≤ . (108)We obtain (cid:12)(cid:12)(cid:12) p [ m ] N ( f ( n N ) | E k ) − p N ( f ( n N ) | θ (cid:48) k ) (cid:12)(cid:12)(cid:12) ≤ N δ , (109)from Eq. (104) by using Lemma 3 and setting a = N δ/ r (cid:48) = N, r = 1.Next, we derive sufficiently large m to bound the right-hand side of Eq. (109) by ε . First, we transform a U ( t ) as a U ( t ) = 1 d Tr [ U ( t )] · Tr [ U ( t )] ∗ (110)= 1 d (cid:88) k e − iE k t · (cid:88) l e iE l t (111)= 1 d (cid:88) k,l e i ( E k − E l ) t (112)= 2 d (cid:88) k>l cos( E k − E l ) t + 1 d . (113)Note that cos x ≥ − x /
2. By using the maximumdifference ∆ max = max k,l | E k − E l | in energy eigenvalues,we have a U ( t ) ≥ − (cid:88) k
0, if m ≥ ∆ t N N − ε , (120)then (cid:12)(cid:12)(cid:12) p [ m ] N ( f ( n N ) | E k ) − p N ( f ( n N ) | θ (cid:48) k ) (cid:12)(cid:12)(cid:12) ≤ ε. (121)3 Estimating the energy scale
We claimed that we can find an upper bound of ∆ max by estimating a U ( t ) for some prior distribution of the sys-tem Hamiltonian H in the sixth section in the main ar-ticle on Quantum algorithmic estimation of the energyscale . In subsection , we present the protocol for es-timating of the energy scale. This protocol employs asubroutine algorithm evaluating a U introduced in Sec. .Evaluation of the failure probability of the protocol isshown in Sec. together with mathematical formulas usedin evaluation. Protocol for estimating of the energy scale
Suppose we have a d -dimensional quantum system for d ≥
3. A precise description first involves regarding H asa random variable. We denote the minimum and maxi-mum energy eigenvalue of H by E min and E max , respec-tively, which are both a random variable themselves. Let F ( E min = e min , E max = e max ) be the probability densityfunction (pdf) of the random variables. We shall alsouse a shorthand notation F ( E min , E max ), when there isno fear of confusion, and likewise for other random vari-ables. We denote the k × k identity matrix by I k .The prior distribution of H shall be such that F ( E min = e min , E max = e max )= (cid:40) F (∆ max = e max − e min ) ( e max ≥ e min )0 (otherwise) , (122)where ∆ max := E max − E min (123)is to be regarded as a random variable.Each Hamiltonian has d energy eigenvalues E , . . . , E d .We set E = E min and E d = E max . Given a set of specificbounds ( E min , E max ) = ( e min , e max ), it remains to specifyother d − F ( E k | E min , E max ) of each eigenvalue E k is independent of other eigenvalues, i.e., F ( E , . . . , E d − | E min , E max )= Π d − k =2 F ( E k | E min , E max ) , (124)and each F ( E k | E min , E max ) is a uniform distribution overthe range from E min to E max .A straightforward argument to obtain a reliable esti-mate of the upper bound is to choose a value E UB suffi-ciently large so that the probability p ( E UB ≥ ∆ max ≥
0) = (cid:90) E UB F (∆ max = x ) dx (125) is sufficiently large. Such E UB is an upper bound for∆ max since ∆ max ≤ E UB . This estimate, however, maybe too conservative, if F (∆ max ) has a very broad distri-bution. A tighter estimation is possible by quantum pro-cess tomography, but as already discussed, this requiresan exponentially increasing time cost. The evaluation of a U ( t ) , which we describe next, provides a more efficientestimation of ∆ max .The estimation protocol on the upper bound of ∆ max is as follows:1. Choose numbers t > (cid:15) so that 1 − (cid:0) d + d − dπ (cid:1) ≥ (cid:15) > j = 0 and c = d + d − dπ + (cid:15) .3. Evaluate a U ( t/ j ) .4. If a U ( t/ j ) < c , change j as j + 1 and go back toStep 3, otherwise go to Step 4.5. Conclude that ∆ max ≤ j · π/t .Suppose that the protocol terminates with j = J . Theprobability p J, fail that this estimation procedure fails isgiven by the probability of choosing a Hamiltonian suchthat satisfies a U ( t/ j ) < c for j from 1 to J − a U ( t/ J ) ≥ c , but simultaneously ∆ max ≥ J · π/t , i.e., p J, fail = p (cid:0) ∆ max t/ J ≥ π,a U ( t ) < c, . . . , a U ( t/ J − ) < c, a U ( t/ J ) ≥ c (cid:1) , (126)where each a U ( t/ j ) is used as a random variable. We willsee that p J, fail ≤ (cid:18) − d(cid:15) (cid:15) (cid:19) Φ( d, (cid:15) ) , (127)where Φ( d, (cid:15) ) := (cid:18) d (cid:15) log(1 + d(cid:15) ) (cid:19) . (128) Evaluation of a U The subroutine algorithm to evaluate a U is given asfollows (Fig. 6). First, we prepare a probe consisting of a d -dimensional ancilla and control qubit. The ancilla andtarget are set to the completely mixed state I d /d and thecontrol qubit to the state | (cid:105) . The initialization of thetarget can be achieved, for instance, by first swappingthe state of the ancilla with the target, after which wereset the ancilla to I d /d .Next, we apply the Hadamard gate H on the controlqubit. This yields the state | + (cid:105)(cid:104) + | ⊗ I d d ⊗ I d d , controlancillasystem I/d W U(t) |0 〉 I/d H H {0,1} FIG. 6. Quantum circuit representation of the algorithm toevaluate the coherence factor a U ( t ) . W U ( t ) represents the clas-sically conditioned quantum gate. A quantum circuit repre-sentation of W U ( t ) is given by Fig. 4(a). where the terms in the tensor product correspond to thecontrol qubit, ancilla, and target, respectively. We per-form the classically conditioned quantum gate W U . Theresulting state is12 d · I ⊗ I d ⊗ I d + 14 d · (cid:20) σ z ⊗ (cid:0) U ⊗ U † + U † ⊗ U (cid:1) − iσ y ⊗ (cid:0) U ⊗ U † − U † ⊗ U (cid:1) (cid:21) . (129)Another Hadamard gate is applied on the control qubit.The reduced density matrix of the control qubit becomes˜ ρ c = I | Tr [ U ] | · σ z (cid:16) I + a U ( t ) σ z (cid:17) . (130)Thus the average value obtained from the measurementof σ z on the control qubit isTr[˜ ρ c σ z ] = a U ( t ) . (131)Therefore, M iterations of this algorithm estimate a U ( t ) with error of O (cid:16) / √ M (cid:17) , which decreases independentlyof dimension. Hereafter, we assume that we can obtain a U ( t ) with sufficiently high accuracy. Probability of incorrect estimation
In this subsection, we evaluate the failure probability ofthe energy scale estimation protocol given by Eq. (127).It is clear that for any two random variables X and Y , p ( X ≥ x, Y ≥ y ) ≤ p ( X ≥ x ) . (132)Thus, we have a bound on p J, fail , i.e., p J, fail ≤ p (cid:0) ∆ max t/ J ≥ π, a U ( t/ J ) ≥ c (cid:1) . (133)Thus, it suffices to prove that p (cid:0) ∆ max t ≥ π, a U ( t ) ≥ c (cid:1) ≤ (cid:18) − d(cid:15) (cid:15) (cid:19) Φ( d, (cid:15) )(134) holds for any t > a U ( t ) as a random variable. Observe that p (cid:0) ∆ max t ≥ π, a U ( t ) ≥ c (cid:1) = (cid:90) ∞ π p ( a U ( t ) ≥ c (cid:48) | t ∆ max ) F ( t ∆ max = x ) dx. (135)We introduce pdf of a U ( t ) and t ∆ max , i.e., F (cid:0) a U ( t ) , t ∆ max (cid:1) . (136)For any given ∆ max , the probability of obtaining a U ( t ) ≥ c for any number c is given by p (cid:0) a U ( t ) ≥ c | t ∆ max (cid:1) = (cid:90) ∞ c F ( a U ( t ) = y | t ∆ max ) dy, (137)where F ( a U ( t ) | t ∆ max ) is the conditional probability den-sity, F ( a U ( t ) | t ∆ max ) = F ( a U ( t ) , t ∆ max ) F ( t ∆ max ) . (138)We first analyze an average property of a U ( t ) for a given t ∆ max . By definition, a U ( t ) is the magnitude of a complexnumber γ U ( t ) = (cid:104) (cid:80) dk =1 exp( − iE k t ) (cid:105) /d . Recall that theconditional density function F ( E , . . . , E d − | E min , E max )is given by Eq. (124), where each E k is a random variablewith an independent and identical distribution. Thus theaverage (cid:104) γ U ( t ) (cid:105) is (cid:104) γ U ( t ) (cid:105) = 1 d d (cid:88) k =1 (cid:104) e − iE k t (cid:105) . (139)For each k = 2 , . . . , d − F ( E k | E min , E max ) is a uni-form distribution over the range between E min and E max ,which implies (cid:104) e − iE k t (cid:105) = e − iE max t − e − iE min t − it ∆ max . (140)We see that the r.h.s. is independent of k . This showsthat (cid:12)(cid:12) (cid:104) γ U ( t ) (cid:105) (cid:12)(cid:12) ≤ d + d − d · t ∆ max , (141)where we have used the triangle inequality on the numer-ator and that (cid:12)(cid:12) (cid:104) e − iE t (cid:105) (cid:12)(cid:12) = (cid:12)(cid:12) (cid:104) e − iE d t (cid:105) (cid:12)(cid:12) = 1 . (142)While this only shows an average behavior of γ U ( t ) , thefollowing extension of Bernstein’s inequality asserts thatthe average is a good representation of the whole for anysum of independent random variables.5 Lemma 4 (Bernstein’s inequality (for vectors) [23]) . Let { X , . . . , X d } be a set of d independent (not necessarilyidentical) random variables that return an n -dimensionalvector in C n and S be S := (cid:80) dk =1 X k . Given that foreach X k , || X k − (cid:104) X k (cid:105)|| ≤ , then for any ˜ t > p (cid:0) || S − (cid:104) S (cid:105)|| ≥ ˜ t (cid:1) ≤ (cid:18) − ˜ t / σ + (˜ t/ (cid:19) (cid:32) t log (cid:0) t/σ ) (cid:1) (cid:33) , (143) where σ ≡ (cid:80) dk =1 (cid:104)|| X k || (cid:105) − ||(cid:104) X k (cid:105)|| . Crudely speaking, it states that the sum of independentrandom variables distributes most likely around the av-erage of the sum. This is a particular instance of “con-centration of measure” phenomena known in probabilitytheory [24].Hence, it is unlikely that a U ( t ) , which is equal to (cid:12)(cid:12) γ U ( t ) (cid:12)(cid:12) , deviates far from (cid:12)(cid:12) (cid:104) γ U ( t ) (cid:105) (cid:12)(cid:12) . Formally, we ar-rive at the following bound on conditional probability p ( a U ( t ) ≥ c (cid:48) | t ∆ max ), Theorem 2.
For any (cid:15) > , if t ∆ max ≥ π , then p (cid:18) a U ( t ) ≥ d + d − dπ + (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) t ∆ max (cid:19) ≤ (cid:18) − d(cid:15) (cid:15) (cid:19) (cid:18) d (cid:15) log(1 + d(cid:15) ) (cid:19) . (144) Proof.
By Eq. (141), (cid:12)(cid:12) (cid:104) γ U ( t ) (cid:105) (cid:12)(cid:12) ≤ d + d − dπ . (145)Therefore, for any Hamiltonian such that a U ( t ) ≥ d + d − dπ + (cid:15), (146)it must be that (cid:12)(cid:12) γ U ( t ) − (cid:104) γ U ( t ) (cid:105) (cid:12)(cid:12) ≥ (cid:15). (147)This implies that p (cid:18) a U ( t ) ≥ d + d − dπ + (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) t ∆ max (cid:19) ≤ p (cid:0)(cid:12)(cid:12) γ U ( t ) − (cid:104) γ U ( t ) (cid:105) (cid:12)(cid:12) ≥ (cid:15) (cid:12)(cid:12) t ∆ max (cid:1) . (148)Next, we apply Bernstein’s inequality. To do so, we re-gard a complex number as a 2-dimensional vector andtake X k = e − iE k t S = d · γ U ( t ) d (cid:88) k =1 X k . (150) Note that pdf for each k satisfies F (cid:0) X k = e − iE k t / (cid:1) = F ( E k | E min , E max ) , (151)where we take X and X d as a constant random variable.It is easy to see that X k satisfy the necessary conditionsto apply Bernstein’s inequality. Let ˜ t in Eq. (143) be˜ t = d(cid:15) , (152)since p (cid:0) (cid:12)(cid:12) γ U ( t ) − (cid:104) γ U ( t ) (cid:105) (cid:12)(cid:12) ≥ (cid:15) (cid:12)(cid:12) t ∆ max (cid:1) = p (cid:18) || S − (cid:104) S (cid:105)|| ≥ d(cid:15) (cid:12)(cid:12)(cid:12)(cid:12) t ∆ max (cid:19) . (153)Notice that the r.h.s. of Eq. (143) is monotonically in-creasing with respect to σ , which satisfies σ ≤ d (cid:88) k =1 (cid:104)|| X k || (cid:105) = d (cid:88) k =1 (cid:28) (cid:12)(cid:12)(cid:12)(cid:12) e − iE k t (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) = d . (154)Therefore, we obtain the desired bound given by p (cid:18) a U ( t ) ≥ d + d − dπ + (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) t ∆ max (cid:19) ≤ (cid:18) − d(cid:15) (cid:15) (cid:19) (cid:18) d (cid:15) log(1 + d(cid:15) ) (cid:19) . (155) [1] V.B. Braginsky, Y.I. Vorntsov, and K.S. Thorne, Science , 547 (1980).[2] S. Mukamel,
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