Quantum Information and Computation, Vol. 1, No. 0 (2001) 000–000

©Rinton Press

SPEED-UP AND ENTANGLEMENT IN QUANTUM SEARCHING

Samuel L. Braunstein and Arun K. Pati

Informatics, Bangor University, Bangor, LL57 1UT, UK

and

Institute of Physics, Sainik School Post

Bhubaneswar-751005, Orissa, India.

Received (received date)

Revised (revised date)

We investigate the issue of speed-up and the necessity of entanglement in Grover’s quantum search algorithm. We find that in a pure state implementation of Grover’s algorithm entanglement is present even though the initial and target states are product states. In pseudo-pure state implementations, the separability of the states involved defines an entanglement boundary in terms of a bound on the purity parameter. Using this bound we investigate the necessity of entanglement in quantum searching for these pseudo-pure state implementations. If every active molecule involved in the ensemble is ‘charged for’ then in existing machines speed-up without entanglement is not possible.

Keywords: Computational speed-up, Quantum entanglement, Grover’s algorithm, Pseudo-pure states

Communicated by: to be filled by the Editorial

## 1 Introduction

It was Feynman’s insight to realize that quantum systems cannot be efficiently simulated on conventional classical computers . Subsequently, Deustch suggested that if one could build computers exploiting the principles of quantum theory one might be able to speed up the computation process compared to classical approaches . Quantum computers aim to make use of quantum interference and entanglement between different parts of a bulk quantum system to give such an essential difference .

In recent years an important question has been: What will it take for quantum computers to surpass their conventional classical counterparts in speed and efficiency? The answer consists of algorithms and their efficient implementation. The first promising algorithms for quantum computers were discovered by Deutsch and Jozsa for function testing , by Shor for factoring  and by Grover for searching . While the efficiency of these algorithms is today well established, the conditions for achieving quantum efficiencies have been the subject of recent controversy . Indeed, experimental implementations of Grover’s algorithm on pseudo-pure state machines using liquid-state Nuclear Magnetic Resonance (NMR) [8, 9] have been claimed to already achieve quantum efficiencies [10, 11, 12, 13] in spite of their apparent inability to have produced entanglement to date . Similarly, the Deutsch-Jozsa algorithm has been implemented on NMR quantum computers [15, 16].

The aim of this paper is to argue that the original version of Grover’s algorithm  on multiple qubits necessarily involves quantum entanglement, even though the initial and target states are product states. Further, by counting each active molecule as contributing to the computational resources for pseudo-pure state machines, we show in a non-asymptotic analysis that not only is entanglement necessary to achieve a speed-up in quantum searching, but it must be present throughout the computation. If one uses a different resource counting method there may be speed-up without entanglement. Thus, we have resolved the role played by entanglement in quantum searching on one whole class of quantum computers. This non-asymptotic result unequivocally proves that a class of liquid-state NMR machines [10, 11, 12, 13] cannot be used for performing faster-than-classical quantum computation.

## 2 Efficiency in Quantum Computation

Typically, efficiency is quantified by relative ‘speed’ or how the number of steps needed to complete an algorithm scales with the size of the ‘input’ the algorithm is fed. Two ubiquitous ‘exponential’ problems are searching and factoring: All known algorithms for solving them on conventional computers scale roughly exponentially with input size (e.g., the length of the list to be searched or size of the number to be factored). Discoveries of ‘fast’ quantum algorithms [4, 5, 6] set new bounds on computational goals and standards.

In order to determine the efficiency of an algorithm on a quantum computer, the conventional measure of ‘speed’ must first be re-evaluated. Clearly, for the scaling behavior to be a sensible measure of efficiency that may be used to compare the performance of very different kinds of computers, there must be no ‘hidden costs’ that grow in an unreasonable manner (i.e., faster than the scaling itself). For example, any increase in the size of the computer itself, or number of resources it utilizes relative to the input, should not exceed the scaling of the number of steps.

Pseudo-pure state quantum computers are welcome candidates for such re-valuations. Indeed, such machines have been implemented using liquid-state NMR and have been proposed for a wide variety of quantum algorithms [8, 9, 15, 16, 17] and their efficiency has already been determined for asymptotically large systems (i.e., many qubits) for Shor’s factoring algorithm. It was found that, in this asymptotic limit, an absence of entanglement would lead to an exponential decrease in the probability for obtaining the correct answer . Thus, if Shor’s algorithm were to be implemented on such machines, an exponentially large number of resources would be required to boost this low probability. Because Shor’s algorithm provides an exponential speed-up, it may not be so surprising that the weirdest features of quantum mechanics, i.e., entanglement, are required for its implementation.

An important caveat of this analysis stems from the extreme unlikelihood of constructing a sufficiently large machine that could be subjected to such asymptotic analysis. For example, it was observed that for liquid-state NMR implementations the signal scales as with increasing numbers of qubits (i.e., active spins) . This inherent scaling problem suggests that regardless of entanglement or speed-up there seems little hope of ever reaching the asymptotic regime.

The situation is very different for Grover’s search algorithm. First, Grover’s quantum algorithm provides a much more modest quadratic (as opposed to exponential) speed-up over any search on a conventional computer; thus one might expect it to be more robust with respect to a loss of entanglement. Secondly, there have now been several experiments demonstrating this algorithm on small (few-qubit) NMR machines and claiming quantum efficiencies [10, 11, 12, 13]. For these experiments, the asymptotic signal scaling mentioned above is not relevant, and the entanglement question thus becomes the primary consideration in assessing speed-up.

In fact, speed-up in Grover’s algorithm can be evaluated quite naturally in terms of query complexity. The query complexity formulation yields a non-asymptotic result which may be applied to any size problem. We argue that, in pure-state as well as pseudo-pure state implementations, if only separable (i.e., unentangled) states are accessed then the speed-up predicted by Grover’s algorithm fails to materialize. There exists a peculiar exception (though one not accessible to existing pseudo-pure state machines) for a search space of size four where entanglement is not necessary. Since generalizations of this interesting exception are as yet unknown, the heuristic claim that entanglement is necessary for a scalable quantum computation implementing Grover’s algorithm still holds. Our conclusion is also consistent with the analysis of Grover’s algorithm, implemented on a device exploiting only superposition, but at the cost of scalability due to exponentially growing resources . It should be noted that in addition to the necessary condition we analyze the sufficient condition for speed-up of quantum searching on pseudo-pure state machines.

The outline of the paper is as follows. In section 3, we begin with a straight-forward demonstration that the pure-state version of Grover’s algorithm involves entanglement during the quantum searching operation. In section 4, we analyze the quantum search that can be implemented on pseudo-pure states and derive a bound on the purity parameter for the pseudo-pure states to be separable. In section 5, to determine the necessity of entanglement in the pseudo-pure state version, two complementary criteria are derived: one for the presence of entanglement (as a function of the number of qubits) and one for the query complexity of the algorithm, yielding a measure of speed-up relative to the classical algorithm. We impose these two criteria in the few-qubit regime, one at a time, and obtain a one-to-one relation between entanglement and speed-up, showing both that there can be no speed-up without entanglement, and, conversely, that in case a speed-up is in fact achieved, entanglement must be present throughout the computation. In section 6, we discuss how pseudo-pure states are not good candidates for mimicking pure-state dynamics. Finally, we discuss the implications of our results.

## 3 Quantum searching with pure states and entanglement

According to the standard formulation of the search problem , we are given an unknown binary function , which returns for a unique ‘target’ value and otherwise, over the domain , with . Our goal is to find such that . In Grover’s algorithm, the inputs are mapped onto the states of quantum bits (qubits) such as spin- particles. The quantum problem thus becomes one of maximizing the overlap between the state of these qubits and target state . This is equivalent to maximizing the probability of obtaining the desired state upon measurement. The initial state of these qubits is taken to be an equal superposition of all possible bit strings, i.e.,

 ⏐Ψ0⟩=1√NN−1∑x=0⏐x⟩. (1)

The Grover operator defined as is used repeatedly in the algorithm, where , with being the target (ideally the final) state and is the Hadamard transformation. Thus, the Grover operator corresponds to a small rotation in the two-dimensional subspace spanned by the initial and target states. Each such rotation requires a single evaluation of . Thus, unlike a classical search, the quantum search monotonically rotates the state towards the target.

Let us start by considering the pure-state version of Grover’s algorithm. After iterations of the Grover operator the combined qubit state (1) evolves to 

 ⏐Ψk⟩=cosθk√N−1∑x≠y⏐x⟩+sinθk⏐y⟩, (2)

where and satisfies . The search is complete when which takes iterations of the Grover operator and hence this many evaluations of the function .

We will now show that although the initial and target states are product states the intermediate states through which system evolves are entangled. Since these states are superpositions of product states they are expected to be entangled. But how much entanglement is there in these intermediate states? This is difficult to quantify, as we do not have a proper measure of entanglement for quantum systems consisting of an arbitrary number of subsystems. However, we can consider the system as being bipartite, with one subsystem consisting of a single qubit and the second subsystem all the rest. In this way, we will be able to quantify the bipartite entanglement by calculating the reduced density matrix of any single qubit. We then ask how close this reduced state is to a maximally entangled qubit (using say the Hilbert-Schmidt norm criterion).

The reduced density matrix of the th qubit is

 ρℓ(k) = tr1,2,..,ℓ−1,ℓ+1,..n(⏐Ψk⟩⟨Ψk⏐) (3) = a2kH⏐0⟩⟨0⏐H+b2k⏐jℓ⟩⟨jℓ⏐+akbk√N(2⏐jℓ⟩⟨jℓ⏐+⏐~jℓ⟩⟨jℓ⏐+⏐jℓ⟩⟨~jℓ⏐),

where , and the single bit , . Without loss of generality we take and the density matrix can be expressed in standard form, i.e.,

 ρℓ(k)=12[\small 1\kern-3.8pt\normalsize 1+→s(k)⋅→σ]=[1−s(k)]\small 1\kern-% 3.8pt\normalsize 12+s(k)P, (4)

where , and is a pure state projector. The components of the Bloch vector after iterations are

 sx(k) = N−2N−1cos2θk+1√N−1sin2θk sy(k) = 0 sz(k) = 1N−1cos2θk−sin2θk. (5)

The bipartite entanglement in the pure state may be characterized by calculating the von Neumann entropy of this reduced state. Using the expansion formula

 log(1−s2\small 1\kern-3.8pt\normalsize 1+sP)=log(1−s2)+Plog(1+2s1−s), (6)

which holds for any , the von Neumann entropy may be calculated to be given by

 S[ρℓ(k)] = −tr[ρℓ(k)logρℓ(k)] (7) = 1−1−s(k)2log[1−s(k)]−1+s(k)2log[1+s(k)].

The right-hand-side of this expression is independent of the choice of the remaining qubit . Therefore, (7) holds for any one qubit versus qubit partioning. It shows that the reduced density matrix of the single qubit does not arise from a maximally entangled state of qubits, as the von Neumann entropy is not exactly unity. Since the reduced state of Eq. (3) is not pure the full state must be entangled. To see how impure the state in Eq. (3) is one may calculate the linear entropy of it which is given by

 L[ρℓ(k)]=tr[ρℓ(k)−ρℓ(k)2]=1−s(k)22. (8)

If the linear entropy is zero the state is pure and as it approaches the state approaches a completely random mixture. In the quantum search algorithm the parameter can never be zero because that would mean that and are simultaneously zero, which cannot be satisfied. So although the reduced density matrix of the qubit may lie close to the completely mixed state it can never become the identity one.

We now calculate the Hilbert-Schimdt norm of the difference of two density matrices: namely, the completely mixed one and the reduced state given by Eq. (3). This will yield a measure of closeness to the completely mixed state. This Hilbert-Schmidt distance for th iteration during quantum search algorithm is given by

 d(k)2 = ∥\small 1\kern-3.8pt\normalsize 12−ρℓ(k)∥2HS=tr[\small 1% \kern-3.8pt\normalsize 12−ρℓ(k)]2 (9) = 12−L[ρℓ(k)]=s(k)22.

The distance provides an idea of how the reduced state of an individual qubit behaves during the th iteration. It shows that the reduced density matrix of the qubit differs from a completely random mixture by an order of . One can see from Eq. (5) and (9) that for and for the reduced density matrix of any remaining qubit is pure, implying that the whole state must have been non-entangled. Thus, we see that although the initial and target states are separable, the intermediate states through which the system evolves are always entangled.

## 4 Quantum searching with pseudo-pure state and entanglement

In the following we discuss the issue of entanglement in quantum searching with pseudo-pure states. These states naturally arise in liquid-state NMR machines. Here one faces the difficulty of accessing a pure state because the system is in thermal equilibrium at room temperature. Instead, one implements Grover’s algorithm on a near random ensemble of molecular spins in a liquid, with a small preference for the spins to point along an external magnetic field; the size of this preference is quantified by the purity parameter [typically ] [8, 9, 10, 11, 12, 13]. For a sufficiently low spin polarization (corresponding to a sufficiently low purity parameter), the system can be well-approximated by a pseudo-pure state representation described by

 ρ=1−ϵN\small 1\kern-3.8pt\normalsize 1N+ϵ⏐Ψ⟩⟨Ψ⏐, (10)

where is the identity matrix of dimension . (In the high regime, an exact description can be obtained by reverting to the full Boltzmann distribution.) Since, for experimentally feasible values of (now and in the foreseeable future), the pseudo-pure state description suffices, we henceforth restrict our discussion to them.

Let us now consider quantum searching on pseudo-pure quantum states. After iterations of the Grover search operator, one obtains the state

 ρk=1−ϵN\small 1\kern-3.8pt\normalsize 1% N+ϵ⏐Ψk⟩⟨Ψk⏐, (11)

where is given by Eq. (2)

Let us first look at the diagonal form of the reduced state of any remaining qubit (denoted by the index ) in the pure state given in Eq. (5). This has positive eigenvalues , which are independent of but depend on the iteration index . These eigenvalues sum to one and their product, after some calculation, is given by

 λ1(k)λ2(k)=N(N−2)2(N−1)2sin2(2kθ0)cos2θk. (12)

This allows us to decompose the full state at step into a Schmidt basis

 ⏐Ψk⟩=√λ1(k)⏐g′⟩⏐e⟩−√λ2(k)⏐e′⟩⏐g⟩, (13)

where describes an orthonormal basis for the th qubit and is a pair of Hilbert space vectors for the other qubits. This expression will help us to derive a bound on the purity parameter.

In order to study the entanglement present in the pseudo-pure state given by Eq. (11) let us project it onto the 4-dimensional subspace spanned by the set , , , . The resulting 4-dimensional density matrix is

 ρ(4)k=N4+ϵ(N−4)(1−ϵN\small 1\kern-3.8pt\normalsize 14+ϵ⏐Ψk⟩⟨Ψk⏐). (14)

The boundary between separability and entanglement for such 4-dimensional states may be specified in terms of the so-called fidelity . The states (14) are separable when and are entangled otherwise , where . Since projection onto a subspace cannot create entanglement, it follows from this condition that if the original unprojected state (11) is separable, then we must have

 ϵ≤ϵk≡11+N√λ1(k)λ2(k). (15)

Similarly, the original state (at stage of the search) is entangled whenever . Similar bounds have been earlier derived when the pure state part was a maximally entangled state . However, the present bound quantifies the separability region of pseudo-pure states for each iteration during Grover’s algorithm. Thus one can tell at each stage of computation whether the state is entangled for a given value of purity parameter.

## 5 Speed-up and Entanglement for Pseudo-pure states

Let us now return to a careful re-evaluation of what it means for a pseudo-pure state implementation of Grover’s algorithm to demonstrate a speed-up compared to a classical search on a conventional computer. First, we note that the important common element in either a quantum or a classical search algorithm is the repeated need of evaluating the function . Thus, we shall restrict our comparison to its number of evaluations. This is formally known as studying the query complexity of the problem. Second, since Grover’s algorithm is probabilistic, there is no upper-bound to the worst-case number of evaluations. Instead, we study the expected number of evaluations for finding the target. To be fair then, we must compare this with the expected number of function-evaluations on a conventional computer. If we exclude the use of an exponentially growing auxiliary memory to store failed trials then this classical query complexity is

 Nclass=(N+2)(N−1)N (16)

for finding a single object out of entries. This value may be achieved by systematically stepping through the possible locations for the object and evaluating to see if the object is there. If it is not found in this way by step then we would know it is at the final location . (The specific value for classical query complexity quoted in Ref.  for can be obtained from our general result above).

How does this compare to the expected number of function evaluations for quantum search on pseudo-pure state implementations? The probability of finding the target state after iterations is

 p(k)=⟨y⏐ρk⏐y⟩=[1+ϵ(Nsin2θk−1)]N<1. (17)

This probability must be amplified statistically through repetitions or parallelism. The expected number of repetitions required to identify the target is just the reciprocal of this probability. Each such repetition involves function-evaluations for each run of the algorithm, plus one to test the result. In order to give Grover’s algorithm its maximal advantage, we shall optimize the speed (rather than the probability). Thus, the optimal expected number of function evaluations is just

 Npseudo=mink(k+1)p(k)≡(kopt+1)p(kopt), (18)

where is the optimal number of iterations of the Grover operator. Thus, a pseudo-pure state quantum computer can search faster than a conventional computer, provided

 Npseudo

In the above analysis we have utilized the standard projective measurement which is a desirable operation in any quantum information processing unit. However, the liquid-state NMR machines use weak measurements and the success probability obtained in a strong measurement is always greater than the success probability for a weak measurement. So in our analysis we give extra benefit to the experimental situation in tolerating the error. In any case, if we take weak measurement the situation for NMR machines will be no better (as that will increase the number ).

We have now developed the tools to effectively answer the question: Can Grover’s algorithm yield a speed-up on any existing pseudo-pure state implementations [10, 11, 12, 13]? Since all liquid-state NMR experiments performed so far have generated only separable ensembles , with purity parameters that fit well our pseudo-pure state treatment above, let us see what effect separability has on efficiency. From Eq. (15) separability throughout the search implies . Thus, separability places an upper bound on the purity parameter (see values for this bound in Ref. ) and hence a lower bound, , on the quantum query complexity. This lower bound is given in Table 1 for the optimized Grover algorithm for search spaces up to qubits. (The trend we find continues for arbitrarily large numbers of qubits.) The surprising observation is that, for more than qubits, the optimal quantum strategy is not to use the liquid-state NMR machine at all and simply guess the answer (equivalent to sampling the initial random state). Clearly, this is never as good as a systematic conventional search.

Table 1

Curiously, we note that our table shows a speed-up even without entanglement for the qubit implementation. That this algorithm requires no entanglement in this case, has also been noted for the pure-state implementation of Grover’s algorithm  (with a similar result for the two qubit implementation of the Deutsch-Jozsa algorithm ). This occurs because the separable target state can be reached by a single application of the Grover operator from the separable initial state; hence the evolution has passed through no entangled states. Without entanglement in the two-qubit case there is also no penalty to the efficiency when enforcing separability through a reduced purity parameter. Despite this curiosity, this speed-up is only possible for unphysically large purity parameters, , where the pseudo-pure description is no longer valid for liquid-state implementations. This observation and the above analysis show that entanglement is necessary for obtaining a speed-up in Grover’s algorithm on a liquid-state NMR machine relative to a classical computer. Let us emphasize that for pure-state machines (e.g., ion-traps) the two-qubit implementation of Grover’s algorithm is realistic and will yield a speed-up of 2.25 (omitting the now superfluous final testing function-evaluation).

As we have seen, the presence of some entanglement is essential to obtain a speed-up, except in the two-qubit case. But how much of it? Let us impose the speed-up condition (19) to obtain a lower bound on the purity parameter within the pseudo-pure state approximation. After iteration , condition (15) implies entanglement is present whenever . We studied this relation numerically for qubits with , and found that entanglement was present after every iteration except possibly the last step of the algorithm when . Thus, we may draw the much stronger conclusion that, for more than two qubits, some degree of entanglement is necessary during the entire computation in order to obtain any speed-up for Grover’s algorithm on a liquid-state NMR machine.

To better understand the repercussions of this result, let us directly address the main objections raised about the role of entanglement in performing real quantum computation in this system. The most dismissive argument has been that quantum computational efficiency derives only from the unitary evolution of quantum states, but is independent of the type of states being used (Laflamme in Ref. ). Now, for the sake of clarity, let us adopt the definition of quantum computing suggested by the same proponent in the same context, namely: Quantum computing consists of efficiently evolving from initial to final density matrices by unitary operations. Since it is not enough to reach the final state probabilistically, we have calculated the query complexity of the problem in detail (assuming perfect unitary evolution). Thus our analysis clearly proves that unitarity alone is not enough to achieve quantum efficiencies.

A different objection correctly points out that liquid-state NMR machines are not exactly described by the pseudo-pure state formalism, and hence the bounds we derive above for entanglement and separability may not be applicable. Indeed, corrections to this bound are necessary in the few-qubit regime, where the bounds for the purity parameter are high. However, in most likelihood, these corrections will not alter our results and may even strengthen them (by raising the lower bound for entanglement). Further, noting that existing liquid-state NMR machines are remarkably far from reaching this range of purity parameters, any possibility of speed-up is quite out of the question.

## 6 Dynamics of Pseudo-pure versus pure states

Granted that entanglement is needed to achieve quantum efficiencies, it has been argued  that the ability of liquid-state NMR machines to physically follow unitary quantum evolution qualifies them as efficient simulators (without speed-up). The reasoning is based on the realization that the observable operators are traceless . Thus, the expectation value of any traceless observable on a pure quantum state is the same, up to a scale factor, as it would be on a pseudo-pure state. This is because for a traceless observable the average in pseudo-pure state is . However, in the quantum world fluctuations play as important a role as expectation values. In general, if an observable is traceless its square need not be. In this case the quantum fluctuations of an operator on a pure state are not equal to those fluctuations on the corresponding pseudo-pure state. For example, if is a traceless operator its root-mean-square fluctuations on pseudo-pure states are determined by

 (ΔΘ)2pseudo=ϵ(ΔΘ)2pure+(1−ϵ)(tr(Θ2)N+ϵ⟨Θ⟩2pure). (20)

As a simple example, consider , where . It is a traceless operator, but its square is not. The uncertainty of the operator on the pure state is zero, whereas on the pseudo-pure state it is

 (21)

In fact, very rarely will it be the case that all the moments (even for the restricted class of traceless operators) are related by simple scale-factors to those produced by a liquid-state NMR quantum computation. Thus, despite the correct expectation values being accessible for traceless operators, we cannot say that liquid-state NMR quantum computers have good dynamics.

## 7 Conclusion and discussion

In conclusion we have shown that the original version of Grover’s algorithm implemented on qubits necessarily generates quantum entanglement during the computation process. Further, we have shown that a quantum computer in a non-entangled pseudo-pure state requires more iterations than a classical computer to perform a virtual database search if one uses our method of counting the number of queries. Thus, for any existing liquid-state NMR set-ups our results preclude any possibility of speed-up in running Grover’s algorithm. This conclusion is based on exact calculations for pseudo-pure state implementations, which show that entanglement is essential throughout the computation. Moreover, even a modest reduction of entanglement is tantamount to a total loss of speed-up. Despite the decisive results reached for pseudo-pure state implementations of this algorithm, we found that entanglement-free speed-up would be possible for the two-qubit case (search space of size four) of Grover’s algorithm on pure-state machines such as ion-trap quantum computers. Our analysis has been with respect to a specific non-asymptotic regime and does not find a way for scalable computation without entanglement. Finally, our analysis does not resolve how other NMR experiments should to be interpreted such as quantum teleportation without entanglement.

Note added: After we completed this work, a recent article has even implemented the quantum search algorithm using only the wave nature of classical Fourier optics  which involves no entanglement! However, one should not be surprised into thinking that this is a contradiction to our findings presented here. What happens with classical devices implementing the search algorithm is that the number of resources needed increases prohibitively with the size of the input.

Acknowledgements: SLB thanks N. Cohen for discussions. AKP gratefully thanks EPSRC for financial support for the year 1998-2000 during which this work was carried out.

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