Quantifying Computational Advantage of Grover's Algorithm with the Trace Speed
aa r X i v : . [ qu a n t - ph ] J a n Quantifying Computational Advantage of Grover’s Algorithm with the Trace Speed
Valentin Gebhart,
1, 2, ∗ Luca Pezz´e, and Augusto Smerzi QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy Universit`a degli Studi di Napoli Federico II, Via Cinthia 21, I-80126 Napoli, Italy (Dated: January 15, 2020)Despite intensive research, the physical origin of the speed-up offered by quantum algorithmsremains mysterious. No general physical quantity, like, for instance, entanglement, can be singledout as the essential useful resource. Here we report a close connection between the trace speedand the quantum speed-up in Grover’s search algorithm implemented with pure and pseudo-purestates. For a noiseless algorithm, we find a one-to-one correspondence between the quantum speed-up and the maximal trace speed that occurs during the algorithm operations. For time-dependentpartial depolarization and for interrupted Grover searches, the speed-up can be bounded by themaximal trace speed. Our results quantify the quantum speed-up with a physical resource that isexperimentally measurable and related to multipartite entanglement and quantum coherence.
Understanding and quantifying the key resource forthe speed-up of quantum computations [1, 2] has beena highly disputed topic over the past few decades [3].There has been particular interest in the role played byentanglement [4–13]. It is known that exponential speed-ups of quantum algorithms implemented with pure statesrequire multipartite entanglement [8, 9]. However, it wasshown that polynomial advantage in query complexitycan be achieved without entanglement [14]. Also, it isan open question whether exponential quantum advan-tage can be reached in mixed-state algorithms in ab-sence of entanglement. Here, other quantum correlationssuch as quantum discord have been indicated as possi-ble candidates for computational resources [3, 15]. Fur-thermore, it was shown that several entanglement mea-sures do not recognize useful entanglement in quantumalgorithms [13]. Other possible resources have been con-sidered such as coherence [16–18], distinguishability [3],contextuality [19], tree size [20] and interference [21].Quantum statistical speeds [22–26] offer a possible ap-proach to quantify useful resources in quantum technol-ogy tasks. As a major example, the quantum Fisherinformation [25, 27], which is the quantum statisticalspeed associated with the Bures distance [25], was shownto fully characterize metrologically useful entanglement[28–30], that is, the entanglement necessary for sub-shot-noise phase estimation sensitivities [31, 32]. One mightconjecture that different statistical speeds may be usefulto characterize the performances of different quantumtasks. Here, we use the trace speed (TS), namely, thestatistical speed associated to the trace distance [1, 23],to quantify the speed-up in Grover’s algorithm [33] inboth absence and presence of dephasing. We show thatin the cases without dephasing, the maximal TS occur-ring during the search algorithm completely determinesthe speed-up. For general pseudo-pure dephasing models[34], we prove that the TS bounds the speed-up, render-ing it a necessary resource for quantum advantage. TheTS is an experimentally relevant measure of quantum co-herence (asymmetry) [35, 36] and witnesses multipartite entanglement [26]. To our knowledge, this is the firstresult for a physical resource in Grover’s algorithm thatgeneralizes to mixed state versions. This can pave theway to a new approach to investigate useful resources inquantum computations.
Grover’s algorithm.
Grover’s search algorithm [33] isone of the most important protocols of quantum com-putation [1, 2]. It searches an unstructured database of N elements for a target ω . The target is marked in thesense that we are given a test function f that vanishesfor all elements but ω . The task is to identify ω with asfew function calls as possible. In the quantum versionof the algorithm, a function call can be used as a mea-surement or as an application of a corresponding unitary,the so-called oracle unitary. As we will discuss shortly,Grover’s algorithm admits a quadratic advantage to clas-sical search algorithms. To utilize the exponential size ofthe dimensionality of composite quantum systems [6], weencode all different elements x of the register into com-putational basis states of n = log N qubits, x ∈ { , } n .Grover’s algorithm is performed by preparing the sys-tem in the register state | ψ in i = 1 / √ n P x | x i , where | x i are the computational basis vectors, followed by k applications of the Grover unitary G = U d U ω . Here,the oracle unitary U ω = 1 − | ω i h ω | represents a func-tion call and the Grover diffusion operator is defined as U d = 2 | ψ in i h ψ in | −
1. After k iterations of the Groverunitary, the state of the system is given by [2] | ψ k i = sin[(2 k + 1) θ ] | ω i + cos[(2 k + 1) θ ] (cid:12)(cid:12) ω ⊥ (cid:11) , (1)where θ = arcsin(1 / √ n ) and (cid:12)(cid:12) ω ⊥ (cid:11) =1 / √ n − P x = ω | x i is the projection of the ini-tial state on the subspace orthogonal to | ω i . Thisyields a probability p k of finding the target state as p k = sin [(2 k + 1) θ ]. After k Gr ≈ ( π/ √ n itera-tions one finds the target state | ω i with probability p k Gr = 1 − O (1 / n ) [2].One defines the cost C for a general search algorithmas the average number of applications of the test function f (or its corresponding oracle unitary) required to findthe target state [34]. Simply counting the oracle applica-tions is also known as query complexity [2], while othercomplexities such as time complexity are usually not con-sidered in Grover’s algorithm (see [6] for a discussion).In the classical search algorithm, the query applica-tion can be thought of as opening one of 2 n boxes, whereeach box represents one state of the register. For an un-structured search algorithm, i.e., in each iteration onerandomly opens one of the 2 n boxes. The average num-ber of steps needed to find the target state is given by C cl = 2 n . If one remembers the outcome of all previoussearches, the cost can be reduced to C cl = 2 n / O (1)[34]. Note that C cl for both structured and unstructuredsearches scales with 2 n .In a quantum search algorithm, one uses k oracle uni-taries and a final oracle measurement yielding the targetwith probability p k , such that the cost is given by [34] C qu ( k ) = k + 1 p k . (2)Hence, the optimal cost is obtained by minimizing C qu ( k )over the number of oracle applications, C qu = min k ( k +1) / ( p k ). Let us emphasize that this definition of the costdoes not distinguish between applying the oracle as aunitary or as a measurement observable.In Grover’s algorithm, the cost function Eq. (2) is notnecessarily minimal for the highest success probability p k of one single search [37]. However, the optimal numberof steps ˜ k Gr and the optimal cost C qu for large n stillscales as ˜ k Gr = r √ n and C qu = K √ n , where r is thesolution of tan(2 r ) = 4 r and K = r/ sin (2 r ), yieldingthe quadratic speed-up over C cl . It was shown that thisspeed-up is optimal [37, 38].Grover’s algorithm can be executed on a single multi-mode system and, therefore, simply makes use of super-position and constructive interference [6, 39, 40]. How-ever, in order to reduce exponential overhead in space,time or energy, one usually considers a system composedof many qubits [6, 39]. In this case, different measures ofbipartite and multipartite entanglement have been usedto detect entanglement during Grover’s algorithm [41–45]. Genuine multipartite entanglement was shown to bepresent already after the first step of the noiseless algo-rithm [41]. However, the quantitative relationship be-tween these measures and speed-up was not resolved. Inparticular, the methods could not be easily applied to anymixed state generalization of Grover’s algorithm. Quan-tum coherence [46–48] and quantum discord [46] havebeen considered as resources in the noiseless algorithmas well. Trace speed.
The TS is the susceptibility of a quantumstate ρ to unitary displacements generated by a genericHamiltonian H [35]. That is, the TS quantifies the dis-tinguishability between ρ and ρ ( t ) = e − iHt ρe iHt for small t . It is defined as [1, 26, 35]TS( ρ, H ) = (cid:13)(cid:13) ∂ t ρ ( t ) (cid:12)(cid:12) t =0 (cid:13)(cid:13) = k [ ρ, H ] k , (3)where [ · , · ] is the commutator and k·k is the l -norm,defined as k A k = tr h √ A † A i for a generic operator A .In general, TS is a measure of coherence, in this caseusually referred to as asymmetry [35]: a state with nocoherence with respect to H , namely a classical mixtureof its eigenstates, will not change under phase displace-ments, while off-diagonal matrix elements (coherences)of ρ are responsible for a finite susceptibility to phasedisplacements. The TS is upper bounded by the quan-tum Fisher information [23]. If the system is a compositesystem of n qubits and H is the sum of local Hamilto-nians H i , H = P ni =1 H i with spec( H i ) = {− / , / } and TS( ρ, H ) > √ nr , it follows that ρ has to be at least( r +1)-partite entangled [26, 29, 30]. Since the value of TSdepends on the generating Hamiltonian H , we considerthe optimization over all Hamiltonians of the above form.When the whole evolution is restricted to the completelysymmetric subspace, it suffices to perform this optimiza-tion over collective spin Hamiltonians, H i = n · σ ( i ) / n is a point on the unit sphere and σ ( i ) are thePauli operators for the i -th qubit. For pure states | ψ i , theoptimized TS coincides with the square root of the largesteigenvalue of the matrix Γ ij = Re[ h J i J j i ] − h J i i h J j i [29].Here, J m = P ni =1 e m · σ ( i ) / e m -direction, m = x, y, z , and h·i is the expectationvalue with respect to the state | ψ i . Pure state algorithm.
Let us first discuss the TS for thestandard version of Grover’s algorithm implemented withpure states and unitary evolution, as introduced above.Without loss of generality, we can take | ω i = | i ⊗ n [49].Since | ψ in i and | i ⊗ n are elements of the completely sym-metric subspace and G commutes with all permutationsof the qubits, the complete evolution is restricted to thesymmetric subspace. This reduces the dimensionality ofthe Hilbert space from 2 n to n + 1, facilitating the com-putation of TS. By neglecting terms in O (1 / n ), one canexactly compute the largest eigenvalue of Γ ij at any step k [50], yielding the optimized TS. In Fig. (1), we showthe optimized TS( k ) for n = 30 qubits. The initiallyseparable state | ψ in i evolves into a multipartite entan-gled state already after the first oracle operation. Mul-tipartite entanglement further increases until reaching amaximal value of TS puremax = r n ( n + 1)2 (4)which occurs at k = k Gr /
2. This detects ( n/ k > k Gr /
2, multipartite entanglement detected bythe TS decreases until the algorithnm reaches the sepa-rable target state | ω i . n k / k Gr T S FIG. 1. The dependence of the optimized trace speed TSon the iteration step k in the pure state Grover’s algorithm(solid line). The dashed lines indicate thresholds abovewhich TS detects bipartite ( √ n ), three-partite ( √ n ) and( n/ p n /
2) entanglement. Here, n = 30, k Gr ≈ ( π/ √ n . Mixed register state.
We now consider Grover’s algo-rithm with the register initialized in a pseudo-pure state,while the algorithm is still implemented with unitary op-erations. For a pure n -qubit state | ψ i , the correspondingpseudo-pure state ρ ψ,ǫ with purity parameter ǫ is definedas ρ ψ,ǫ = ǫ | ψ i h ψ | + 1 − ǫ n I . (5)Pseudo-pure states represent, for small purities, an ap-proximation to the thermal state of the system andtherefore arise naturally in liquid-state NMR (see forinstance Ref. [51]). We replace the pure initial state | ψ in i with the pseudo-pure state ρ ψ in ,ǫ such that, after k Grover iterations, the state of the system is given by ρ k = ǫ | ψ k i h ψ k | + (1 − ǫ ) I / n , with | ψ k i defined in Eq.(1). The probability p k of finding the target state after k steps is p k = ǫ sin ((2 k + 1) θ ) + (1 − ǫ ) / n . Here weobserve that for ǫ = O (1 / n ), it becomes more efficientto just measure the state without any iteration becausethe probability contribution due to the Grover iterationis no longer dominant [7]. However, if ǫ does not de-crease exponentially with n , one can neglect the secondterm in p k . Hence, the minimum of Eq. (2) occurs afterthe same number of steps as in the pure state algorithmwhile its minimal value C qu is simply C qu = C qu , pure /ǫ ,where C qu , pure is the cost of the pure state algorithm.The TS for a pseudo-pure state ρ ψ,ǫ is given byTS( ρ ψ,ǫ , H ) = ǫ TS( | ψ i , H ) because [ H, id] = 0. There-fore, the maximal TS during the algorithm is TS max = ǫ p n ( n + 1) /
2. Hence, for the Grover algorithm with apseudo-pure initial state with purity parameter ǫ , TS de-tects ǫ ( n + 1) / C qu ( n, TS max ) = K √ n TS puremax TS max , (6)where K = r/ sin (2 r ) ≈ .
69 with r being the solutionof tan(2 r ) = 4 r . The quantum speed-up S = C cl /C qu isthus given in terms of TS as S = √ n K TS max TS puremax (7)Note that for ǫ < / ( n + 1), TS does not detect en-tanglement anymore. It was already observed that forpurities ǫ > / n/ the algorithm still offers a speed-up[3, 52, 53], indicating that entanglement detected by TSis not necessary for quantum speed-up. We should em-phasize that the form of Eq. (7) suggests that similarresults can possibly be found connecting the speed-up toother measures of coherence or other quantum proper-ties. Importantly, the choice of the TS was suggestedby the fact that its value for pseudo-pure states has asimple dependence on its pure state value and that italso detects multipartite entanglement for mixed states.We have also considered different quantum statisticalspeeds, other than the TS, such as the generalized quan-tum Fisher information [26]. However, after includingtime-dependent depolarization, the TS proved to havethe closest connection to the speed-up for pseudo-purestate algorithms. Partial depolarization.
The results of pseudo-pure ini-tial states can be generalized to search dynamics sub-ject to time-dependent partial depolarization (see Refs.[54, 55] for earlier investigations). In this case, the stateafter k steps of the algorithm is given by ρ k = ǫ ( k ) | ψ k i h ψ k | + 1 − ǫ ( k )2 n I , (8)where the now time-dependent decreasing purity ǫ ( k )represents both initial impurity and partial depolariza-tion during the algorithm. As can be seen in Fig. (2),different purity functions ǫ ( k ) with the same final pu-rity can lead to different maximal TS during the itera-tion. While the one-to-one correspondence between theTS and the speed-up is generally lost, as shown below,we can still bound the speed-up using TS.For a partial depolarization during the algorithm itturns out that, in general, it is optimal to stop the it-erations and perform the final measurement already atearlier steps k int < ˜ k Gr [54]. Let us first consider in-terrupting the iteration at a time k int ≥ k Gr /
2, i.e., af-ter the pure state algorithm would have already reachedits maximal TS, see Fig. (1). In this case, the costcan be bounded by C qu ≥ K √ n /ǫ ( k int ). This is be-cause if one would completely stop the dephasing, onecould reduce the cost until reaching the optimal value of K √ n /ǫ ( k int ). With ǫ ( k int ) ≤ ǫ ( k Gr /
2) and TS( k Gr / ≤ ϵ f k / k Gr T S P u r i t y ϵ FIG. 2. Purities ǫ ( k ) (solid lines) and trace speeds TS( k )(dashed lines) for an initial pseudo-pure state without dephas-ing (blue), an initial pure state with linearly decaying purity(yellow) and an initial pure state with exponentially decayingpurity (green). Here, n = 30, ǫ f = 0 . k Gr ≈ ( π/ √ n . TS max , one can then bound C qu ≥ K √ n /ǫ ( k Gr / ≥ ( K √ n TS puremax ) / (TS max ), yielding the following bound S ≤ √ n K TS max TS puremax . (9)The case ǫ ( k int ) ≤ ǫ ( k Gr /
2) corresponding to strongdephasing becomes more technical since, in the earlyregime, the maximal TS is not simply bounded by ǫ ( k ) TS puremax . However, as we show in the appendix, thebound Eq. (9) still holds. These results for the case of aninterruption of the iteration due to minimization of thecost can also be applied to the case of a general interrup-tion of the iteration. Stopping the algorithm at any timewill yield an average speed-up which is always boundedby the maximal TS occurring before the interruption. Discussion.
By studying the TS, we have been ableto relate the computational speed of Grover’s algorithmto both multipartite entanglement and quantum coher-ence. It should be noticed that the relation with multi-partite entanglement depends on the n -qubit implemen-tation that we have considered, while the algorithm canalso be implemented with a single 2 n -level system [6]. In-deed, as mentioned above, the operating principle of thealgorithm and the number of queries used (which deter-mines the cost) do not depend on which implementetionwe use. Therefore, multipartite entantanglement cannotbe considered as the key resource for the quantum speed-up. We thus argue that the correct interpretation of ourresult is the evidence that the resource for speed up inquery complexity is quantum coherence as captured bythe TS. However, multipartite entanglement is crucialto reduce other costs such as space or energy [6]. Wepoint out that the interpretation of the TS as quantumcoherence holds for any implementation of the algorithm.The role of quantum coherence during the noiselessGrover’s algorithm has already been investigated in Refs. [46, 47]. These works found a one-to-one correspondencebetween the l -norm of coherence which is decreasingduring the algorithm and the increasing success probabil-ity. Both approaches have not been generalized to mixedstate versions of the algorithm. In our case, a differentmeasure of coherence, namely the TS, is connected tothe average cost of the algorithm. It reaches its maximalvalue during the algorithm and offers a physical resourcealso for pseudo-pure generalizations. In Refs. [46, 47],the l -norm of coherence and the relative entropy of co-herence are used which detect different states as highlycoherent as TS would. For instance, while the l -normdetects the initial state | ψ in i = 1 / √ n P x | x i as maxi-mally coherent, TS would detect ( | i ⊗ n + | i ⊗ n ) / √ ρ (0) and ρ ( t ), for a givenmeasurement observable. A quadratic series expansion ofthe Kolmogorov distance for sufficiently small t yields theKolmogorov speed which is a lower bound to the TS anddepends on the considered measurement observable. TheTS is obtained by maximizing the Kolomogorov speedover all possible observables [1]. Conclusions.
We showed that both in the purestate version of Grover’s search algorithm and a generalpseudo-pure generalization, the trace speed TS can beused to quantify and bound the possible quantum speed-up. These results offer an unprecedented connection be-tween the speed-up in Grover’s algorithm and a physicalresource beyond the case of ideal, noiseless quantum al-gorithms. The TS offers a new and experimentally fea-sible approach to the analysis of quantum advantages.This might inspire further investigations of the still unan-swered search for the origins and quantification of quan-tum advantage. In particular, one could check the impor-tance of the TS and other quantum statistical speeds forother oracle-based quantum algorithms such as, e.g., theDeutsch-Jozsa algorithm or Simon’s algorithm, or gen-eral quantum technology tasks. Also, whether or not theTS is a necessary resource in different noisy variations ofGrover’s algorithm, merits further investigation. 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Smerzi, Witnessingentanglement without entanglement witness operators,Proc. Natl. Acad. Sci. USA , 11459 (2016). APPENDIXProof of the bound for strong dephasing
Let us consider the case of interrupting the searchalgorithm at an early step k ≤ k Gr /
2, i.e., at a step where the pure state algorithm would not have reachedits maximal trace speed TS yet. The cost for stop-ping the algorithm at step k is given by C qu ( k ) =( k +1) / ( ǫ ( k ) sin ((2 k +1) θ )), see Eq. (1). The TS at step k still fulfills TS( k ) = ǫ ( k ) TS pure ( k ), where TS pure ( k ) isthe TS in the pure state algorithm, see Fig. (1). There-fore, we have C qu ( k ) = k + 1sin [(2 k + 1) θ ] TS pure ( k )TS( k ) ≥ k + 1sin [(2 k + 1) θ ] TS pure ( k )TS max = a ( k )TS max (10)where TS max is the maximal TS until the interruptionstep k and we regrouped all other factors into a ( k ). Tofurther examine this expression, we use the exact form ofthe pure state TS, TS pure ( k ), during the algorithm whichis given byTS pure ( k )= r n (cid:16) n − f ( k ) n + p f ( k )] + n [1 − f ( k )] (cid:17) (11)with f ( k ) = cos[4(2 k + 1) /θ ]. We can then compare thefactor a ( k ) with the factor b = K √ n p n ( n + 1) / x = k/ √ n , onefinds for large na ( k ) − b = n √ n √ (2 x ) [ K (cos(4 x ) −
1) + 2 x sin 4 x ] . (12)For 0 ≤ x ≤ π/ ≤ k ≤ π/ √ n = k Gr /
2) and using K ≈ .
69, one finds that a ( k ) − b >
0. Therefore, thebound of Eq. (9) still holds for the regime k ≤ k Gr //