# A complexity measure for continuous time quantum algorithms

###### Abstract

We consider unitary dynamical evolutions on qubits caused by time dependent pair-interaction Hamiltonians and show that the running time of a parallelized two-qubit gate network simulating the evolution is given by the time integral over the chromatic index of the interaction graph. This defines a complexity measure of continuous and discrete quantum algorithms which are in exact one-to-one correspondence. Furthermore we prove a lower bound on the growth of large-scale entanglement depending on the chromatic index.

2

## I Introduction

At the moment, the most popular model of a quantum computer consists of the dimensional Hilbert space of ‘ qubits’ as its memory space, and some one- and two-qubit gates as its set of basic transformations (see e.g. [1]). There are several reasons for taking one- and two-qubit gates as the basic ones: Firstly, from the pure mathematical point of view, it is quite natural to look for a subset of the Lie group of unitary transformations on the Hilbert space generating the whole group. Obviously, one-qubit operations do not generate the whole Lie group, the set of two-qubit gates does [2]. Hence there is no reason for taking more complicated transformations like three-qubit unitary operators as basic ones. Secondly, the model of two-qubit gates might be considered as the attempt to develop quantum computation in strong analogy to the theory of classical devices: Building complex logical networks from two-bit gates is a successful concept of classical computation. The third reason stems from physics. Unfortunately it is only ‘a little bit’ convincing: From the fundamental point of view, particles interact always in the form of pair-interactions, i.e., the total Hamiltonian of the system can be decomposed as

where is a self-adjoint operator acting on the joint Hilbert space of particle and and is the free Hamiltonian of particle (without loss of generality, we can drop the second sum, by reckoning it to the first part). Pair-interactions are the infinitesimal versions of two-qubit gates: every unitary of the form for is a two-qubit gate. On the one hand, this seems to be an important justification for two-qubit gates, since it refers to the form of the fundamental forces of nature, on the other hand, the argument is not really correct: In general there is no obvious correspondence between the time evolution

(1) |

and any finite sequence of two-qubit gates. However, there is an obvious simulation by two-qubit gates in an approximative sense given by the well-known Trotter formula:

(2) |

This example shows, that the simulation of the time evolution caused by a time-independent pair-interaction Hamiltonian might require an infinite number of two qubit gates. Hence it seems to suggest that a definition of complexity based on two-qubit gates does not take into account the most natural form of dynamics of many-particle quantum physics. However, it has been shown [3] that the number of gates required to simulate the right hand side of (2) up to an error is only growing with . Despite the fact that infinite accuracy requires an infinite number of gates, the time for implementing the growing number of unitaries does not tend to infinity if one assumes that the implementation of requires the time (see [3]). Taking this assumption, we will show that the running time of a discrete quantum algorithm for simulating the dynamics given by (1) is determined by the chromatic index of the interaction graph. This turns out to be true even for time-dependent pair-interactions. One might reformulate this result by saying that the running time depends on ‘the complexity of the interaction’. The relevance of this result is twofold: In case realizations of future quantum computers are based on two-qubit gates, it gives exact statements about the complexity of simulating non-autonomous quantum dynamics resulting from pair-interaction Hamiltonians. To our knowledge, this is the first exact analogue between complexity measures of discrete and continuous quantum algorithms. Secondly, our result is relevant for the simulation of arbitrary pair-interaction dynamics by a dynamic with restricted interaction graph: We derive statements about the efficiency for simulating time evolutions based on pair-interactions with high chromatic index by other evolutions with interactions of lower index.

## Ii Discrete and continuous quantum algorithms

In our model of discrete quantum computers, two-qubit gates acting on disjoint pairs of qubits can be implemented simultaneously. We define:

###### Definition 1

A discrete quantum algorithm of depth is a sequence of steps where every step consists of a set of two qubit gates acting on disjoint pairs of qubits. Every step defines a unitary operator by taking the product of all corresponding unitaries in any order. The product is the ‘unitary operator implemented by ’.

The following quantity measures the deviation of a unitary operator from the identity:

###### Definition 2

The angle of an arbitrary unitary operator
is the smallest possible norm^{1}^{1}1Here and in the following denotes the operator norm given by where runs over the unit vectors of the corresponding Hilbert space. of a selfadjoint
operator which satisfies
.
It coincides with the time required for the implementation of if
the norm of the used Hamiltonian is 1.

This term allows us to formulate a modification of the term ‘depth’ which will later turn out to be decisive in connecting complexity measures of discrete and continuous algorithms:

###### Definition 3

Let be the maximal angle of the unitaries performed in step . Then the weighted depth is defined to be the sum .

Assuming that the implementation time of a unitary is proportional to its angle, the weighted depth is the running time of the algorithm. Since this coincidence is based on a possibly unrealistic assumption we will prefer the term ‘weighted depth’ in formulating exact mathematical statements.

Now we want to formalize the notion of quantum algorithms
based on time dependent Hamiltonians. Such
continuous algorithms have already been considered in the
literature^{2}^{2}2In [4] time-dependent Hamiltonian algorithms are called an
‘analog analogue of a digital quantum computer’.
(e.g. see [4, 5, 6, 7, 8, 9]),
but in our approach the Hamiltonians are
explicitly restricted to pair-interactions in many-particle systems:

###### Definition 4

A continuous quantum algorithm of running time is a piecewise Lipschitz continuous function from the interval into the set of those self-adjoint operators acting on which can be decomposed into

where every is a self-adjoint operator acting on the qubits and . Here and the sum runs over the unordered pairs. We say ‘ implements ’ if and is the solution of the non-autonomous differential equation with .

As already mentioned above, the weighted depth of a discrete algorithm for simulating the evolution (1) is finite. More precisely, it is at most if is the number of pairs with the property that . But, if one has additional information about the interactions, one can make further statements about possible parallelization. This is illustrated by the following examples: In case all the pairs with are disjoint, we can perform all the unitaries simultaneously and get the running time . In case of the nearest-neighbor interaction in a one dimensional spin chain, we have only pairs of the form . Hence we can perform all the transformations

simultaneously and the operations

as well. Here, parallelization allows to decrease the running time down to .

We show that the graph theoretical concept of chromatic index offers the appropriate terminology for determining the degree of possible parallelization.

###### Definition 5

[10] The chromatic index of a graph with vertices is the least number of colors required for coloring the edges in such a way that there are no edges with the same color having a common vertex.

For every time during the running time of the continuous algorithm we define a family of undirected graphs:

###### Definition 6

For every non-negative real number and every time we define the interaction graph as follows: Take the qubits as vertices and let the edges be all the pairs with the property .

The following quantity turns out to be decisive for the degree of possible parallelization of the discrete simulation:

###### Definition 7

Let be the chromatic index of . For any time of a continuous algorithm we define the weighted chromatic index as

Furthermore we will need the time integral of :

###### Definition 8

The integrated chromatic index of the quantum algorithm is defined to be

where is the running time of the continuous algorithm.

Actually, the terminology ‘integrated weighted chromatic index’ would be appropriate. We preferred the short terminology.

## Iii Translation between discrete and continuous algorithms

The following theorem suggests that the simulation of an interaction with high integrated chromatic index generically requires a discrete algorithm with high weighted depth. More precisely, it shows that the complexity of a unitary transformation can equivalently be defined as the infimum over the values of the weighted depth of all possible discrete implementations or the infimum over the values of the integrated chromatic index of all possible continuous implementations:

###### Theorem 1

a) Every arbitrary unitary operator acting on n qubits which can be implemented by a discrete quantum algorithm with weighted depth can also be implemented by a continuous quantum algorithm with integrated chromatic index .

b) If there is a continuous algorithm with integrated chromatic index implementing then there is a sequence of discrete algorithms implementing the unitaries such that and the corresponding values of the weighted depth converge to .

Proof:
Statement a) is almost trivial:
Let be a discrete quantum algorithm of depth with the maximal angles
. Let be the set of qubit pairs which are
addressed in the step and the corresponding
set of unitary transformations. Define
a continuous algorithm with running time as follows:
for we^{3}^{3}3Here we denote an interval
by (.) if we do not care about
whether it is open or closed.
define the constant Hamiltonian .
Since the angle of a unitary operator is the norm of ,
we have . Hence,
for every in the interval we have for
and otherwise.
Obviously we have

For statement b) take a partition of into small intervals on which is continuous. Let be one of those intervals. Let

be the positive numbers in its canonical ordering. Furthermore define . Obviously, the function takes constant values on every interval for . Let

be a partition of the set of edges of given by an allowed coloring (in the sense of Definition 5) of the edges corresponding to the chromatic index . For every we proceed as follows: For every define a step of a discrete algorithm by the set of unitaries

Since runs from to we obtain steps for the discrete algorithm. Every step has at most the angle . The weighted depth of these steps is at most . Doing this for every , we obtain a sequence of steps which substitutes the continuous algorithm on the interval up to an error in the order of . The weighted depth of this sequence is at most

(3) |

Without loss of generality we will assume the weighted depth to be equal to the terms in equation (3) since we can blow up the algorithm by transformations which cancel each other out. If the total discrete algorithm is defined by combining the sequences for every interval of the form we get a simulation of the continuous algorithm with an error in the order of . Furthermore the total weighted depth converges to for .

One can consider discrete quantum algorithms as special cases of continuous quantum algorithms in the sense of the proof of part a) of Theorem 1. Then the discrete quantum computer is obtained by restricting the Hamiltonians to those with chromatic index one and norm one for each non-vanishing component . In this sense, part a) of Theorem 1 is a special case of the general principle that continuous time algorithms can be simulated by other continuous time algorithms with lower chromatic indices and the same strength of the pair-interactions where the running time is increased by the quotient of the chromatic indices. We sketch this simple observation in a not too formal way:

At time we have the interaction

where runs over the edges of the graph . For take such that . Take a partition of the edges of into subsets such that the subgraphs with the set of edges have chromatic indices less or equal to . Substitute the small time interval of the original algorithm by the time intervals with in which the Hamiltonian is switched on. The difference of the unitary operators implemented by the substituted algorithm and the original one is of second order in for every subinterval. Hence the total error tends to zero for .

This observation justifies in some sense our point of view that one might think of the chromatic index as the complexity of the interaction. This raises the question whether bounds can be given on the required integrated chromatic index necessary for preparing certain entangled quantum states from an initial product state.

## Iv Complexity bounds for high entanglement

In [11] we gave lower bounds on the depth required for preparing certain states with large-scale entanglement (some of those bounds are easy conclusions from [12]) for large qubit numbers . More specifically, one can show the following: Let be a family of arbitrary self-adjoint operators where every acts on qubit and has operator norm 1. For product states, the variance of the observable

(4) |

grows with . Values of the variance in the order of indicate large-scale entanglement for pure states [13]. In a state obtained by a discrete algorithm with depth we could show [11] the variance to be less or equal to . Hence the emergence of large-scale entanglement in the sense described above requires a depth in the order of .

It is natural to ask whether similar bounds can be shown for the weighted depth and the integrated chromatic index. However, below we present a proof for such a bound. In contrast to the discrete definition of depth the bound is asymptotically independent on in the limit . Whether this is a lack of our estimations or whether it is a hint for a fundamental difference between complexity measures of continuous and discrete algorithms is unclear. We have:

###### Theorem 2

Let be a continuous quantum algorithm with integrated chromatic index implementing the unitary . Let the quantum computer start in the product state . Let be the state obtained by the algorithm. Then for the variance of any observable of the form (4) in the state we have

with

Proof: Due to Theorem 1 we can restrict the proof to the case that is of the same form as that one constructed in the proof of Theorem 1 a), i.e., is for every a sum of pair-interactions acting on disjoint pairs. Furthermore we will assume without loss of generality that the norm of every non-vanishing pair-interaction is one. Then coincides with the running time . We write where is the set of edges of the interaction graph . By solving the differential equation

with we obtain . Explicitly, is given by a Dyson series [14]:

where the integration is carried out over the simplex

Let be a tuple consisting of unordered pairs (which are considered as subsets of ) and one element . Define

Furthermore, for an arbitrary pair of observables we introduce the covariance in the state as

Then the variance of the observable in the state can be written as:

with

where the sum runs over all -tuples of the form with and and -tuples which are defined analogously by using the sets of edges instead of (here runs over all non-negative integers). In the following, we shall sometimes consider an of this form canonically as a subset of and write iff agrees with the rightmost element of or with one of the vertices of at least one for .

Now let the times and be fixed. We define as the set of -tuples with and the property that every has a vertex which is an element of . It is easy to see that implies . The set can be constructed iteratively: Set . Then we have if and only if and and at least one of the vertices of is an element of (here is an informal notation for the -tuple obtained by appending to the -tuple ). In an analogous way we define the set by referring to the sets instead of .

The covariance

(5) |

can only be nonzero if and
have nonempty intersection
(considered as subsets of ).
We show that there are at most

pairs
with this property by proving that
for every there are at most
sets with . This can be seen by
induction over :
For the statement is obvious.
Assume the statement to be true for .
For every
with we have at most elements
of the form since for each vertex
there can be at most one such that
has as a vertex (remember that the corresponding
graph has chromatic index 1).
If the statement can only be true
if . There is at most one
with vertex
. Let be the pair . By assumption there are at most
tuples such that . Hence there are
at most sets with .

Since the norm of every is zero or one by assumption, the norm of can never exceed .

Hence we get

Since the integration is carried out over a simplex of size we obtain:

This power series converges for . Since

we have:

## V Conclusions

We have shown that the running time of a discrete algorithm simulating the evolution defined by a time-dependent Schrödinger-equation with pair-interaction Hamiltonians is given by the time integral over the chromatic index of the interaction graph. This result suggests to take this time integral as a complexity measure for continuous time algorithms and it seems natural to ask for the complexity for the preparation of highly entangled states in many-particle systems. We proved such a bound. At the moment, we cannot decide whether it is tight or not. Given a quantum system with the property that the weighted chromatic index of the relevant interaction Hamiltonian satisfies for every time we conclude that the system requires at least the time in order to produce large-scale entanglement in the sense explained above. If we assume a non-vanishing probability of depolarizing errors for each single qubit, the error probability has to decrease with if large-scale entanglement should be maintained [13]. This strongly suggests that the chromatic index or the strength of the interaction has to increase in the order of .

For nearest-neighbor interactions in solid state physics it is not useful to apply our estimations, since there are considerably tighter bounds (see [15] paragraph 6.2.1) But for mean-field interactions [16], where the weighted chromatic index is already determined by the strength of the pair-interactions, we cannot see any obvious tighter bounds. Hence physical systems like solid states with long-range interactions will present a typical application of our results.

## Acknowledgments

An important part of the ideas presented above have been developed during a discussion with my colleague R. Steinwandt. Part of this work has been supported by the European Union (project Q-ACTA).

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