Geometry of Quantum Computation with Qudits

Abstract

The circuit complexity of quantum qubit system evolution as a primitive problem in quantum computation has been discussed widely. We investigate this problem in terms of qudit system. Using the Riemannian geometry the optimal quantum circuits are equivalent to the geodetic evolutions in specially curved parametrization of SU(dⁿ). And the quantum circuit complexity is explicitly dependent of controllable approximation error bound.

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 e_jk 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, 0_{d − 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 x_jj and complex z_jk, 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 U_d is followed as

with special coefficients α_jk.

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 u_kk 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 U_A ∈ 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 [x_j, x_k] = 0 or c_jkx_s for suitable constants c_jk and generalized Gell-Mann matrix x_s, specially, each generalized Gell-Mann matrix x_s may be generated by the Lie product of two other generalized Gell-Mann matrices x_j and x_k. 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, h_j are real coefficients, i.e., σ_j 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(dⁿ) 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 U₂ ∈ SU(dⁿ) is a unitary matrix generated by H₂(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 H_xM and vertical space V_xM (, g denotes Lie algebra of G). Denote as a local frame of H_qM, 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 M_x.

Proof of lemma 4

The connection and completeness of SU(dⁿ) gives one choice to find the tangent space M_I 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 |h_j| ≤ 1 at time interval [0, T]. Then there is a unitary U_A satisfying

and being synthesized using at most Nd⁴n² one- 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 (e^{i(A + B)t} = e^iAte^iBt + 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 e^iHT 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/(n^kD(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 U_A 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 . U_A 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.

Author Contributions

M.X. proposed the theoretical method. M.X. and X.B. and X.W. wrote the main manuscript text. M.X. and X.B. and Y.X. reviewed the manuscript.

Supplementary Material

Supplementary Information

Geometry of Quantum Computation with Three-qudit systems