Introduction

Quantum algorithms based on the circuit model have been explored to solve difficult problems in terms of classical computers1, e.g., large integer factoring2,3 and quantum searching algorithm4. These quantum models are composed of a sequence of quantum gates which may be generated from unitary evolution operator U(t) with special Hamiltonian. The interacted gates or nonlocal quantum operations are views as the key of circuit complexity. One algorithm is computable or efficient if it can be realized with polynomial interacted gates in terms of system size5,6. One main goal of quantum computation is to investigate efficient quantum circuits to synthesize quantum operation7,8,9.

The quantum circuit complexity for qubit systems has been investigated with different approaches such as the geometric approach10,11,12,13,14. It provides an equivalent statement as the shortest path problem in a curved geometry. The quantum circuit complexity may be reduced to the metric distance between the identity operation and desired unitary using designed Riemannian metric on the unitary space. Thus the metric distance is a good measure of the complexity of synthesizing quantum nonlocal evolution.

In comparison to the qubit system, d-dimensional quantum states (qudits for short) could be more efficient in quantum applications. With larger state space the qudit algorithms may improve channel capacity15,16 and quantum gates implementation17,18,19, increase security20,21,22,23,24 and explore quantum features25,26,27. The qudit systems have also been experimentally realized28,29,30,31. The high-dimensional quantum system maybe provide different quantum correlations and efficient information processing. d degenerate (in the rotating-wave approximation sense) ground states may form the qudit, coherently coupled via a common excited state by pulsed external fields of the same time dependence and the same detuning32. The multiple photon states may also be reformed as qudit with the help of symmetric primitive states under the permutation invariance32,33.

In this paper we focus on the circuit complexity of qudit quantum computations. The results for qubit-systems13 and qutrit-system14 are extended to general qudits. Algebraic methods such as eigenvalue decomposition34,35, cosine-sine decomposition36 or Householder Reflection decomposition37 and qudit computation based on d-level cluster38 have been used to find the optimal qudit circuits. In comparison to these evaluations, the quantum circuit complexity of nonlocal qudit evolutions are revalued in terms of the Riemannian geometry on SU(dn). The qudit system evolution is completely determined by the geodesic equation and the initial value of the Hamiltonian. The minimal geodesics through a desired unitary, will be investigated by deforming the known geodesics of a special metric to the geodesics of the metric of qudit circuits. Thus the qubit or qutrit systems13,14 are improved to qudit ones, in the sense that universal qudit gates are possible to efficiently synthesize a unitary operation with controllable error and present explicit relationship between the quantum circuit complexity and total error bound, which have not ensured in previous results13,14. Three-qudit system as an explicit example is presented in supplementary information.

Results

A quantum gate on n-qudit states is a unitary matrix U ∈ SU(dn) determined by time-dependent Hamiltonian H(t) (with reduced Planck constant) according to the following Schrödinger equation,

with U(0) = I, U(T) = U and . The evolution time t ∈ [0, T]. H can be decomposed with the generalized Gell-Mann matrices. The Lie algebra su(dn) of SU(dn) is different from the Pauli matrices of qubits. Thus we first present necessary results of su(dn).

Let ejk denotes the d-by-d matrix with a 1 in the j, k position and zeros elsewhere, a basis can be described as follows:

Here, diag represents the diagonal matrix, 0d− 2j denotes the zeros of length d − 2j. Thus are traceless and Hermitian matrices which are generalizations of Pauli matrices for qudit systems. These matrices combined with identity matrix span the vector space of d × d Hermitian matrix

with real xjj and complex zjk and are also named with generalized Gell-Mann matrices. They construct one group representation of su(d) and other representations may be obtained by arbitrary unitary transformations. From the unitary matrix representation a recursive parametrization of the unitary matrix Ud is followed as

with special coefficients αjk.

Let with and

be an operator acting on the s-th qudit with and the rest qudits with identity operation I, . The basis of su(dn) is constituted by , , with all possible , where

denotes all operators with generalized Gell-Mann matrices acting on t qudits at sites , respectively and rest with identity. The element in may be named as t-body.

To evaluate the error bound13,14 the norm of an operator A is defined by

which is equivalent to the operator norm given by 〈A, B〉 = trA†B. The norm of generalized Gell-Mann matrices satisfies and with . If we replace these not normalized ukk with , the generalized Gell-Mann matrices are then normal with respect to the operator norm14, still denoted by , is normalized.

Our main result is the following theorem

Main theorem

For any small constant ε, each unitary UA∈ SU(dn) may be synthesized using O(ε−2) one- and two-qudit gates, with error.

To complete the proof necessary lemmas should be proved. Notice that the generalized Gell-Mann matrices have the following communication relations [xj, xk] = 0 or cjkxs for suitable constants cjk and generalized Gell-Mann matrix xs, specially, each generalized Gell-Mann matrix xs may be generated by the Lie product of two other generalized Gell-Mann matrices xj and xk. By using the tensor product , it easily follows that . Thus all 3-body interaction may be generated by 1-body and 2-body. By induction it is not difficult to prove the following lemma.

Lemma 1

The 1-body and 2-body are universal for SU(dn).

This lemma may be reformed as: all s-body items (s ≥ 3) in the basis of su(dn) can be generated by the Lie bracket products of 1-body and 2-body items. With the lemma only problem is to address the approximate accuracy under the limited 2-body items.

The time-dependent Hamiltonian H(t) is represented as

where denotes all possible one and two-body interactions, denotes all other more-body interactions, hj are real coefficients, i.e., σj denotes all possible , defined in equation (8) and denotes all possible defined in equation (8). The cost of unitary operation U synthesization with Hamiltonian is defined as, similar to the qubit case,

where p > 0 is the penalty paid for three-and more-body items13,14. This cost gives rise to a natural distance in SU(dn). A curve between the identity operation I and U is a smooth function,

The curve length is given by . is invariant under different parameterizations of , letting F(H(t)) = 1 by rescaling H(t) the evolution time . The distance D(I, U) between I and U is defined by

The function F(H(·)) has defined a norm associated to a right invariant Riemannian metric13 whose metric tensor is presented as

With this metric the distance D(I, U) equals to the shortest path of the geodesic equation . Here 〈·, ·〉 is the inner product on the tangent space su(dn) and J is an arbitrary operator in su(dn). The Lemma 1 with the basis of su(dn) shows all q-body items (q ≥ 3) may be generated by Lie bracket products of 1-body and 2-body items. Thus the metric in equation (14) is reasonable to find the minimal length solution to the geodesic equation13, because the multiple-body items will be ignorable for large p. The one- and two-body items mainly contribute to the solution.

Based on the statement above the Hamiltonian H(t) may be projected onto one- and two-qudit items H2(t). The projection error is controllable in terms of large penalty p. And then the evolution according to H2(t) is divided into small time intervals and approximated by a constant mean Hamiltonian over each interval. Moreover, the evolution according to the constant mean Hamiltonian over each interval may be composed of a sequence of one- and two-qudit quantum gates13,14. The main problem is to evaluate the total errors introduced by these approximations.

Lemma 2

If U ∈ SU(d n ) is generated by H(t) satisfying ||H(t)|| ≤ c in time interval [0, T], then

where is the mean Hamiltonian.

Proof

The unitary operator U(t) has defined one Dyson operator38 with the evolution |ψ(t)〉 = U(t)|ψ(0)〉. Its Tomonaga-Schrödinger equation is defined

with and U(T) = U. This representation leads to the Neumann series of U(T) as

with . The second term is . Choosing for an integer L, then because of . Thus we have

for L → ∞, from the standard norm inequality ||XY|| ≤ ||X||||Y|| and the condition ||H(t)|| ≤ c.

Lemma 3

Suppose U 2 ∈ SU(d n ) is a unitary matrix generated by H 2 (t) with evolution time T. Then

Proof

Assume two time-dependent Hamiltonians H(t) and (with reduced Planck constant) generate U(t) and V(t) respectively

with U(0) = V(0) = I, U(T) = U and V(T) = V. Using the triangle inequality and unitary invariance of operator norm, we obtain

Here, is used. The Euclidean norm of is given by . From the Cauchy-Schwartz inequality it follows

Therefore from equations (21) and (22) the distance

Lemma 4

The distance D(I, U) always has a supremum independent of p in terms of large p.

To prove this lemma, the Chow's theorem39 is required. Let M be a connected manifold and a connection on a principal G-bundle. The Chow's Theorem states that the tangent space at any point q ∈ M with the horizontal space HxM and vertical space VxM (, g denotes Lie algebra of G). Denote as a local frame of HqM, then any two points on M can be connected by a horizontal curve if iterated Lie brackets of evaluated at q span the tangent space Mx.

Proof of lemma 4

The connection and completeness of SU(dn) gives one choice to find the tangent space MI as its Lie algebra su(dn) at the identity matrix I. Thus I and U ∈ SU(dn) is connected by a unique geodesic link associated the Riemannian metric in equation (14). The distance D(I, U) is monotonically increased in the penalty p. Moreover, from the Lemma 1 su(dn) may be generated with iterated Lie brackets of 1-body and 2-body items in the basis . With the Chow's Theorem the curve connecting I and U in horizontal subspace is unique in terms of the metric in equation (14) because of the invariance of su(dn). Otherwise, special geodesic with initial tangent vector in su(dn) may be existed. Hence D(I, U) has an optimal upper bound d0 independent of p.

Lemma 514

If A and B are two unitary matrices, then

for any integer k.

Based the unitary invariance this lemma is easily followed from

Lemma 6

Suppose H(t) only contains one- and two-body Hamiltonian with |hj| ≤ 1 at time interval [0, T]. Then there is a unitary UAsatisfying

and being synthesized using at most Nd4n2one- and two-qudit gates, where c is a constant and N is large integer.

Proof

Divide the interval [0, T] into N intervals of size T/N. In every interval, define a unitary matrix

where L = d2(d2 − 1)(n2 − n)/2 = O(n2) (d2 − 1 generators of su(dn)) denotes the total number of possible one- and two-body interactions in H(t). From the Trotter formula40 (ei(A + B)t = eiAteiBt + O(t2) for two Hermitian matrices A and B) there exists a constant c such that

Using the Lemma 5

which means that one can approximate eiHT with at most N L ≤ Nd4n2 quantum gates. The error is controlled by the division number N.

With these lemmas we can prove main Theorem.

Discussion

The present Theorem have explained the circuits complexity of in quantum computation with n-qudit systems in terms of the Riemannian geometry. Similar to the qubit case12,13 and qutrit case14 the optimal quantum circuit is reduced to the shortest path problem based on special curved geometry of SU(dn). The qudit systems present different algebraic derivations from qubits and qutrits. Especially to realize the unitary invariance of the norm we take use of operator norm in equation (9) similar to the qutrit case while the norm ||M||1 = max〈ϕ|ϕ〉 = 1{|〈ϕ|M|ϕ〉|} used before is not unitary invariant12,13. For instance, consider

O denotes zero matrix. One has ||M||1 = 1/2 and . Generally, one has ||M||1 ≤ ||M|| from the Cauchy-Schwartz inequality. The equal case ||M||1 = ||M|| is derived from M†M = I or M† = M.

Moreover, the main theorem is more efficient than previous results12,13,14. Our result shows that the approximation error for synthesizing quantum qudit operation can be close to zero in terms of nonlocal quantum gate cost. However, it12,13 reads

which is inexplicit because D(I, U) depends p and the sum of the last two terms may be lower bounded with nonzero constant by choosing δ = 1/(n2D(I, U)). Therefore the approximation interval length δ should be smaller such as 1/(nkD(I, U)) with k > 3. Moreover, their error bounds have not shown explicit relationship between the approximation error and total nonlocal gate cost. Our result presents that the number of approximate nonlocal quantum operations is the order of ε−2 with the approximation error bound ε. In comparison to the previous results34,35,36,37,38, the present geometric way shows the relationship between the quantum approximation and the evolution of special geodesic equation. With the quantum circuit model the detailed circuit has to be found for generating U while one needs to find the shortest geodesic curve linking I and U for geometric method. So, these results may be considered as different quantum approximations with their own features.

Methods

The Lemmas 2–6 are used to prove Main Theorem. Let H(t) (at time [0, T]) be the time-dependent normalized Hamiltonian generating the minimal geodesic of length d(I, U) = T. Let H2(t) be the projected Hamiltonian with only the one- and two-body items and generating U2 with evolution time T. From the Lemma 3 it follows that

Divide the time interval [0, T] into N equal parts with length δ = T/N. Let be the unitary generated by H2(t) with evolution time δ in the j-th time interval and be the unitary generated by the mean Hamiltonian with evolution time δ. because with reasonable scaling. Hence where L = d2(d2 − 1)(n2 − n)/2 is the number of one- and two-body items in H(t), i.e. the number of terms in H2(t). Then from the Lemma 2 it follows that

Moreover, from the Lemma 6 there exists a unitary synthesized using at most Nd4n2 one- and two-qudit gates and satisfies

with bounded on every time interval and c = d2n.

In the follow we show that how to construct U2 and UA in terms of and , respectively. In fact, U2 is generated using of as

with and defined on [(j − 1)δ, jδ] for j = 1,···, N. Letting be an identity operator on time interval [0, Nδ] except [(j − 1)δ, jδ]. is generated by the Hamiltonians . UA can be generated similarly. Therefore

From equations (33), (34) and (36) the approximation error

for a constant c = dnd0 + c0d0d4n2 + c1n2, where and c0 is a constant. Choose a suitable penalty p so that D(I, U) = T satisfies 8d0/9 ≤ T ≤ d0 from the Lemma 4 and p−1/2 ≤ ε and small satisfying δ ≤ ε. Since may be synthesized with Nd4n2 one- and two-body gates the total number is N2d4n2 = O(ε−2) one- and two-body gates.