Abstract
Determining the quantum circuit complexity of a unitary operation is an important problem in quantum computation. By using the mathematical techniques of Riemannian geometry, we investigate the efficient quantum circuits in quantum computation with n qutrits. We show that the optimal quantum circuits are essentially equivalent to the shortest path between two points in a certain curved geometry of SU(3^{n}). As an example, threequtrit systems are investigated in detail.
Introduction
Due to the quantum parallelism, quantum computers can solve efficiently problems that are considered intractable on classical computers^{1}, e.g., algorithm for finding the prime factors of an integer^{2,3} and quantum searching algorithm^{4}. A quantum computation can be described as a sequence of quantum gates, which determines a unitary evolution U performed by the computer. An algorithm is said to be efficient if the number of gates required grows only polynomially with the size of the problem. A central problem of quantum computation is to find efficient quantum circuits to synthesize desired unitary operation U used in such quantum algorithms.
A geometric approach to investigate such quantum circuit complexity for qubit systems has been developed in^{5,6,7}. It is shown that the quantum circuit complexity of a unitary operation is closely related to the problem of finding minimal length paths in a particular curved geometry. The main idea is to introduce a Riemannian metric on the space of nqubit unitary operations, chosen in such a way that the metric distance d(I, U) between the identity operation and a desired unitary U is equivalent to the number of quantum gates required to synthesize U under certain constraints. Hence the distance d(I, U) is a good measure of the difficulty of synthesizing U.
In fact, ddimensional quantum states (qudits) could be more efficient than qubits in quantum information processing such as key distribution in the presence of several eavesdroppers. They offer advantages such as increased security in a range of quantum information protocols^{8,9,10,11,12}, greater channel capacity for quantum communication^{13}, novel fundamental tests of quantum mechanics^{14}, and more efficient quantum gates^{15}. In particular, hybrid qubitqutrit system has been extensively studied and already experimentally realized^{16,17}. The higher dimensional version of qubits provides deeper insights in the nature of quantum correlations and can be accessed by encoding qudits in the frequency modes of photon pairs produced by continuous parametric downconversion.
In particular, the threedimensional quantum states, qutrits are of special significance. For instance, in the stateindependent experimental tests of quantum contextuality, three ground states of the trapped ^{171}Yb^{+} ion are mapped to a qutrit system and quantum operations are carried out by applying microwaves resonant to the qutrit transition frequencies^{18}. The solidstate system, nitrogenvacancy center in diamond, can be also served as a qutrit system, in which the electronic spin can be individually addressed, optically polarized, manipulated and measured with optical and microwave excitation. Due to its long coherence time, it is one of the most promising solid state systems as quantum information processors.
In this paper we study the quantum information processing on qutrit systems. We generalize the results for qubitsystems^{7} to qutrit ones. The efficient quantum circuits in quantum computation with n qutrits are investigated in terms of the geometry of SU(3^{n}). Threequtrit systems are investigated in detail. Compared with the results for qubit systems^{7}, our results are more fined, in the sense that by using enough one and twoqutrit gates it is possible to synthesize a unitary operation with sufficient accuracy. While from^{7}, it is not guaranteed that the error of the approximation would be arbitrary small.
Results
A quantum gate on nqutrit states is a unitary operator U ∈ SU(3^{n}) determined by timedependent Hamiltonian H(t) according to the Schrödinger equation, For qutrit case the Hamiltonian H can be expanded in terms of the GellMann matrices. As the algebra related to the nqutrit space has rather different properties from the qubits case in which the evolved Pauli matrices have very nice algebraic relations, we first present some needed results about the algebra su(3^{n}).
Let λ_{i}, i = 1, …, 8, denote the GellMann matrices,
Let be an operator acting on the αth qutrit with λ_{k} and the rest qutrits with identity I. The basis of su(3^{n}) is constituted by {Λ_{s}}, s = 1, …, n, where 1 ≤ α_{1} < α_{2} < … < α_{s} ≤ n, 1 ≤ k_{i} ≤ 8. Λ_{s} stands for all operators acting on s qutrits at sites α_{1}, α_{2}, …, α_{s} with GellMann matrices , , …, respectively, and the rest with identity. We call an element in {Λ_{s}} an sbody one. By using the commutation relations among the GellMann matrices, it is not difficult to prove the following conclusion:
Lemma 1 All sbody items (s ≥ 3) in the basis of su(3^{n}) can be generated by the Lie bracket products of 1body and 2body items.
In the following the operator norm of an operator A will be defined by which is equivalent to the operator norm given by <A, B> = tr A^{†}B. The norm of above GellMann matrices satisfies λ_{i} = 1, i = 1, …, 7, and . If we replace λ_{8} with , the GellMann matrices are then normal with respect to the definition of the operator norm, and the basis of su(3^{n}), still denoted by {Λ_{s}}, is normalized.
A general unitary operator U ∈ SU(3^{n}) on nqutrit states can be expressed as U = U_{1}U_{2}…U_{k} for some integer k. According to Lemma 1, every U_{i} acts nontrivially only on one or two vector components of a quantum state vector, corresponding to a Hamiltonian H_{i} containing only one and twobody items in {Λ_{s}}, s = 1, 2.
The timedependent Hamiltonian H(t) can be expressed as where: (1) in the first sum , σ ranges over all possible one and twobody interactions; (2) in the second sum , σ ranges over all other morebody interactions; (3) the h_{σ} are real coefficients. We define the measure of the cost of applying a particular Hamiltonian in synthesizing a desired unitary operation U, similar to the qubit case, where p is the penalty paid for applying three and morebody items.
Eq. (3) gives rise to a natural notion of distance in the space SU(3^{n}) of nqutrit unitary operators with unit determinant. A curve [U] between the identity operation I and the desired operation U is a smooth function, The length of this curve is given by . As d([U]) is invariant with respect to different parameterizations of [U], one can always set F(H(t)) = 1 by rescaling H(t), and hence U is generated at the time t_{f} = d([U]). The distance d(I, U) between I and U is defined by The function F(H) can be thought of as the norm associated to a right invariant Riemannian metric whose metric tensor g has components: These components are written with respect to a basis for local tangent space corresponding to the coefficients h_{σ}. The distance d(I, U) is equal to the minimal length solution to the geodesic equation, 〈dH/dt, J〉 = i〈H, [H, J]〉. Here 〈·, ·〉 is the inner product on the tangent space su(3^{n}) defined by the above metric components, and J is an arbitrary operator in su(3^{n}).
From Lemma 1 in the basis {Λ_{s}} of su(3^{n}), all the qbody items (q ≥ 3) can be generated by Lie bracket products of 1body and 2body items. To find the minimal length solution to the geodesic equation, it is reasonable to choose such metric (6), because the influence of there and morebody items will be ignorable for sufficiently large p. It is the one and twobody items that mainly contribute to the minimal geodesic.
We first project the Hamiltonian H(t) onto H_{P}(t) which contains only one and twoqutrit items. By choosing the penalty p large enough we can ensure that the error in this approximation is small. We then divide the evolution according to H_{P}(t) into small time intervals and approximate with a constant mean Hamiltonian over each interval. We approximate evolution according to the constant mean Hamiltonian over each interval by a sequence of one and twoqutrit quantum gates. We show that the total errors introduced by these approximations can be made arbitrarily smaller than any desired constant.
Let M be a connected manifold and a connection on a principal Gbundle. The Chow's theorem^{19} says that the tangent space M_{q} at any point q ∈ M can be divided into two parts, the horizontal space H_{q}M and the vertical space V_{q}M, where and ( denotes the Lie algebra of G). Let be a local frame of H_{q}M. Then any two points on M can be joined by a horizontal curve if the iterated Lie brackets evaluated at q span the tangent space M_{q}.
Lemma 2 Let p be the penalty paid for applying three and morebody items. If one chooses p to be sufficiently large, the distance d(I, U) always has a supremum which is independent of p.
Proof
As SU(3^{n}) is a connected and complete manifold, the tangent space at the identity element I can be looked upon as the Lie algebra su(3^{n}). For a given right invariant Riemannian metric (6), there exists a unique geodesic joining I and some point U ∈ SU(3^{n}). With the increase of p, the distance d(I, U, p) of the geodesic joining I and U ∈ SU(3^{n}) increases monotonically.
On the other hand, according to Lemma 1, 1body and 2body items in the basis {Λ_{s}} can span the whole space su(3^{n}) in terms of the Lie bracket iterations. Under the metric Eq.(6), from the Chow's theorem we have that the horizontal curve joining I and U ∈ SU(3^{n}) is unique, since the subspace spanned by 1body and 2body items is invariable. Or there exists such a geodesic that its initial tangent vector lies in the subspace spanned by 1body and 2body items. Hence the distance d(I, U, p) has a sup d_{0} which is independent of p.
Lemma 3 Let H_{P}(t) be the projected Hamiltonian containing only one and twobody items, obtained from a Hamiltonian H(t) generating a unitary operator U, and U_{P} the corresponding unitary operator generated by H_{P}(t). Then where  ·  is the operator norm defined by (2), and p is the penalty parameter in (6).
Proof
Let U and V be unitary operators generated by the timedependent Hamiltonians H(t) and J(t) respectively,
By integrating above two equations in the interval [0, T], we have where U(T) = U, V(T) = V and U(0) = V(0) = I have been taken into account.
Since we have
Using the triangle inequality and the unitarity of the operator norm  · , we obtain:
The Euclidean norm of the Hamiltonian is given by . From the CauchySchwarz inequality, we have 1Moreover, if H contains only three and morebody items, we have
Therefore which gives rise to (7).
Remark
From Lemma 3, by choosing p sufficiently large, say p = 9^{n}, we can ensure that U − U_{P} ≤ d([U])/3^{n}. Moreover, since the distance d(I, U) is defined by d(I, U) = min_{µ[U]}d[U], Lemma 3 also implies that U − U_{P} ≤ d(I, U)/3^{n}.
Lemma 4 If U is an nqutrit unitary operator generated by H(t) satisfying H(t) ≤ c in a time interval [0, Δ], then where is the mean Hamiltonian.
Proof
Recall the Dyson series^{20}:
We choose t_{i} ≤ Δ/(i + 1) and set the first term in the above series to be I. Hence the second term is . We have where we have used the standard norm inequality XY ≤ X Y, the condition H(t) ≤ c, and .
Proposition 1 If A and B are two unitary operators, then
Proof
We begin with N = 2. It is easy to verify that
Now suppose that this inequality holds for N − 1, N ≥ 3, i.e., A^{N}^{−1} − B^{N}^{−1} ≤ (N − 1)A − B. Then for N we have
Lemma 5 Suppose H is an nqutrit one and twobody Hamiltonian whose coefficients satisfy h_{σ} ≤ 1. Then there is a unitary operator U_{A} which satisfies and can be synthesized by using at most c_{1}n^{2}/Δ one and twotribit gates, where c_{1} and c_{2} are constants.
Proof
We need a modified version of the Trotter formula^{1}: let A and B be Hermitian operators, then e^{i}^{(A+B)Δt} = e^{iA}^{Δt}e^{iB}^{Δt} + O(Δt^{2}). We divide the interval [0, Δ] into N = 1/Δ intervals of size Δ^{2}. In every interval, we define a unitary operator
There are L = 32n^{2} − 24n = O(n^{2}) one and twobody items in H. From the modified Trotter formula, there exists a constant c_{2} such that
By using Proposition 1, we have
It means that one can approximate e^{−iHΔ} by using at most Nc_{1}n^{2} = c_{1}n^{2}/Δ quantum gates for some constant c_{1}.
From the above we have our main result:
Theorem 1 Using O(n^{K}d(I, U)^{3}) () one and twoqutrit gates it is possible to synthesize a unitary U_{A} satisfying U − U_{A} ≤ c, where c is any constant.
Theorem 1 shows that the optimal way of generating a unitary operator in SU(3^{n}) is to go along the minimal geodesic curve connecting I and U. As an detailed example, we study the threequtrit systems. In this case the right invariant Riemannian metric (6) turns out to be a more general one^{21}, where , p is the penalty parameter and s is the parameter meaning that onebody Hamiltonians may be applied for free when it is very small, maps the threequtrit Hamiltonian to the subspace containing only onebody items, to the subspace containing only twobody items, and to the subspace containing only threebody items. According to the properties of the GellMann matrices, they satisfy , , .
Set , , and . From the geodesic equation , where , we have which gives rise to the solution where S(0) = S_{0}, T(0) = T_{0} and Q(0) = Q_{0}.
The corresponding Hamiltonian has the form: . According to the assumption 〈H(t), H(t)〉 = 1 for all time t, we have , , and . The term in H(t) is of order p^{−1/2}, and hence can be neglected in the large p limit, with an error of order tp^{−1/2}. Also the term containing p^{−1} in the exponentials of T can be neglected with an error at most of order t^{2}(s^{1/2}p^{−1} + p^{−1/2}). Therefore one can define an approximate Hamiltonian
The corresponding solution of the Schrödinger equation satisfies
Denote . Then and . Thus we have
Generally one can expect that S_{0} + Q_{0} is much lager than T_{0}, and S_{0} + Q_{0} is nondegenerate. can be simplified at the firstorder perturbation, where denotes the diagonal matrix by removing all the offdiagonal entries from T_{0} in the eigenbasis of S_{0} + Q_{0}. Therefore we see that it is possible to synthesize a unitary satisfying , where c is any constant, say c = 1/10.
Discussion
We have investigated the efficient quantum circuits in quantum computation with n qutrits in terms of Riemannian geometry. We have shown that the optimal quantum circuits are essentially equivalent to the shortest path between two points in a certain curved geometry of SU(3^{n}), similar to the qubit case where the geodesic in SU(2^{n}) is involved^{7}. As an example, threequtrit systems have been investigated in detail. Some algebraic derivations involved for qutrit systems are rather different from the ones in qubit systems. In particular, we used (2) as the norm of operators. The operator norm of M used in^{7} is defined by M_{1} = max_{〈ψψ〉 = 1}{〈ψMψ〉}, which is not unitary invariant in the sense that M_{1} = U M_{1} = M U_{1} is not always true for any unitary operator U. For instance, consider and . One has M_{1} = 1/2. However, . Generally, from CauchySchwarz inequality one has M_{1} ≤ M. If M^{†}M = I or M^{†} = M, then M_{1} = M.
Moreover, the final results we obtained are finer than the ones in^{7}. Our result shows that if k in formula is taken to be sufficiently large, U − U_{A} can be sufficiently small. However, the approximation error estimation in^{7} reads
First, since d(I, U) is dependent of the penalty parameter p, there should exist a pindependent bound to guarantee that 2^{n}d(I, U)/p is small for sufficiently large p. Second, if one chooses Δ as scale 1/n^{2}d(I, U), the sum of the last two terms of the right hand side is 9/2 + c_{2}/d(I, U) + O. Therefore the scale should be smaller, for example, 1/n^{k}d(I, U) and k > 3. As Δ takes the scale of 1/n^{2}d(I, U) in^{7}, it can not guarantee that the error in the approximation could be arbitrary small.
Due to the special properties of the Pauli matrices involved in qubit systems, many derivations for qubit systems are different from the ones for qutrit systems. Nevertheless, the derivations for qutrit systems in this paper can be generalized to general high dimensional qudit systems.
Methods
In deriving Theorem 1, we use Lemmas 25. Let H(t) be the timedependent normalized Hamiltonian generating the minimal geodesic of length d(I, U). Let H_{P}(t) be the projected Hamiltonian which contains only the one and twobody items in H(t) and generates U_{P}. According to Lemma 3, they satisfy Divide the time interval [0, d(I, U)] into N parts with each of length Δ = d(I, U)/N. Let be the unitary operator generated by H_{P}(t) in the jth time interval, and be the unitary operator generated by the mean Hamiltonian . Then using Lemma 4 and inequality we have As F(H) is scaled to be one, , one has . Hence where L = 32n^{2} – 24n is the number of one and twobody items in H(t), i.e. the number of the terms in H_{P}(t)).
Applying Lemma 5 to on every time interval, we have that there exists a unitary which can be synthesized by using at most c_{1}n^{2}/Δ one and twoqutrit gates, and satisfies U_{P} and U_{A} can be generated in terms of and , respectively. We show how to generate U_{P} by use of below. First, can be generated by : with . The unitary operator generated by satisfies which can be transformed into with , where is constant in [Δ, 2Δ]. At last we have generated by the Hamiltonians . U_{A} can be generated similarly.
Therefore
From (10), (11) and (12) we obtain: where and c_{0} is a constant.
As mentioned in Lemma 2, the distance d(I, U) has a sup d_{0} for sufficiently large p. For example, we choose a suitable penalty p so that d(I, U, p) satisfies . If we choose Δ to be sufficiently small, e.g. with k sufficiently large, U – U_{A} will be sufficiently small, As we need c_{1}n^{2}/Δ one and twobody gates to synthesize every , we ultimately need one and twobody gates.
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Acknowledgements
The work is supported by NSFC under number 11275131.
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School of Mathematical Sciences, Capital Normal University, Beijing 100037, P. R. China
 Bin Li
 , ZuHuan Yu
 & ShaoMing Fei
School of Mathematics and Statistics, Northeast Normal University, Jilin, Changchun 130024, P. R. China
 Bin Li
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B.L. and Z.H. and S.M. wrote the main manuscript text. All authors reviewed the manuscript.
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The authors declare no competing financial interests.
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Correspondence to ShaoMing Fei.
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