Quantum Resource Estimates

Chapter III: Low-Rank Approximation of Electronic Structure Hamiltonian . 45

3.5 Quantum Resource Estimates

is given by Eq. (3.25), where𝑉 = 𝐻−𝐻⁰andΨ⁽𝑛⁰⁾ =𝜓⁰

0. If𝜓⁰

0is accurate enough, then the correlation energy (defined as 𝐸⁰

𝑐 = 𝐸⁰− 𝐸⁰

𝐻 𝐹, where 𝐸⁰

𝐻 𝐹 is the ground state energy obtained using Hartree-Fock) with the first order correction is accurate toO (𝜀²).

Next, let us consider the implementation of the basis rotations. As written, the single-particle basis changes𝑈^(ℓ) can be implemented using ^𝑁₂

− ^{𝑁−𝜌ℓ}

2

Givens rotations [48]. (Details are given later at the end of this section.) These rotations can be implemented efficiently using two-qubit gates on a linearly connected architec- ture [1, 2]. If one takes𝑆_𝑍 spin symmetry into account, then one can perform basis rotations separately for spin-up and spin-down orbitals, requiring 2 ^𝑁/₂²

- 2 ^(𝑁−𝜌₂^ℓ)/2 Givens rotations and a circuit depth (on a linear architecture) of (𝑁+𝜌_ℓ)/2.

Lastly, using a fermionic swap network, the component corresponding to evolution of the pairwise operator𝑉^(ℓ) can be implemented in ^𝜌₂^ℓ

linear nearest-neighbor two-qubit gates, with a two-qubit gate depth of exactly 𝜌_ℓ [1].

Altogether, the gate count for implementing the full Trotterized exponentiated op- erator is ^𝑁₂

+Í

ℓ 𝜇

𝑁 2

− ^{𝑁−𝜌ℓ , 𝜇}

2 + ^{𝜌ℓ , 𝜇}

2

for both the Hamiltonian𝐻and the uCC cluster operator𝜏. Note that the sum overℓstarts from 1 and the initial basis rotation is written explicitly. For𝐻, the indices𝜇are dummy indices and can be ignored.

To realize this algorithm on a near-term device, where the critical cost model is the number of two-qubit gates, one can implement the gates directly in hardware [49], which requiresÍ^𝐿

ℓ=1

𝑁 𝜌_ℓ 4 + ^𝜌2ℓ

4 −𝜌_ℓ

gates on a linear nearest neighbor architecture, with circuit depthÍ^𝐿

ℓ=1 𝑁 2 + ^3𝜌ℓ

2 . If decomposing into a standard two-qubit gate set (e.g. CZ or CNOT), the gate count would be three times the above count.

From our analysis of the decomposition steps, we expect the Trotterized Hamiltonian exponential to have a gate count of 𝑁_{𝑔𝑎𝑡 𝑒 𝑠} ∼ O (𝑁³) for fixed molecular size and increasing basis size. (We expect to see O (𝑁²) scaling for asymptotically large molecular sizes, but none of the systems studied here reach that limit.) Furthermore, the divergence from linear scaling we saw earlier with increasing alkane chain length in hℓiis no longer as visible due to tails in the distribution in 𝜌_ℓ). We expect the Trotterized uCC exponential to have a gate count of 𝑁_{𝑔𝑎𝑡 𝑒 𝑠} ∼ O (𝑁⁴) for increasing molecular size andO (𝑁³) with increasing basis size. These scalings are verified in Fig. 3.4, which plots the two-body gate counts for Trotterized𝐻⁰and𝜏⁰expontentials with different truncation thresholds.

For a system size of about 50 qubits (believed to roughly be the quantum computer size at which quantum supremacy can be demonstrated [50]), a single Trotter step can be implemented in roughly 4,000 layers of parallel gates, assuming a linear architecture.

Partial basis rotation

Here we will walk through implementing a partial basis rotation, which can be implemented using ^𝑁₂

− ^{𝑁−𝜌ℓ}

2

Givens rotations.

Recall that the single particle basis rotation𝑈^(ℓ) is defined such that

˜ 𝑎^†

𝑝 =

𝑁

Õ

𝑞=1

𝑈_{𝑝 𝑞}^(ℓ)𝑎^†

𝑞 𝑎˜_𝑝 =

𝑁

Õ

𝑞=1

𝑈^(ℓ)_{𝑝 𝑞}

∗

𝑎_𝑞 . (3.27)

According to the Thouless theorem [51], one can perform basis rotations of fermionic operators on quantum computers by applying a series of rotations that act on two rows (𝑝,𝑞) of𝑈^(ℓ) at a time. These rotations are determined by the angles of the Givens rotations 𝑟_{𝑝 𝑞}(𝜃_{𝑝 𝑞}) used to perform a QR decomposition of𝑈^(ℓ). Because 𝑈^(ℓ) is unitary, the rotations will yield a diagonal 𝑅 where the diagonal elements are of magnitude 1, and the QR decomposition only requires performing rotations on the ^𝑁₂

elements below the diagonal. The number of two-body gates used to build the unitary operator is the same as the number of Givens rotations, and can be performed in linear depth on a device with linear connectivity when performed in a particular order [1]. (It is straightforward to extend this to basis rotations of bosonic systems. In fact, the necessary rotations to build the unitary operator would more closely reflect the Given’s rotations.)

Here, we are interested in performing an approximation basis transformation us- ing O (𝜌_ℓ𝑁) Givens rotations, where 𝜌_ℓ ≤ 𝑁 and is the number of eigenvalues retained after making the eigenvalue truncation low-rank approximation. Consider the eigenvalue decomposition of the auxiliary matrix used to obtain Eq. (3.17),

𝑁

Õ

𝑞=1

L𝑝 𝑞^(ℓ)𝑈^(ℓ)

𝑞𝑖 =𝜆^(ℓ)

𝑖 𝑈^(ℓ)

𝑝𝑖 , (3.28)

whereL^(ℓ) is theℓ-th auxiliary vector reshaped into an 𝑁× 𝑁 square matrix, and 𝜆^(ℓ) is the diagonal matrix of the corresponding eigenvalues ordered by decreasing magnitude.

In the eigenvalue truncation of L^(ℓ), we only use the 𝜌_ℓ largest eigenvalues or, equivalently, the eigenvectors associated with the first 𝜌_ℓ columns of 𝑈^(ℓ). This means the sizes of the rotation matrices are reduced, and the eigenvalue equation becomes

¯ 𝑈^(ℓ)

†

L^(ℓ)𝑈¯^(ℓ) =𝜆¯^(ℓ)

𝑖

, (3.29)

where ¯𝑈 and ¯𝜆 are matrices comprising the first 𝜌_ℓ columns of 𝑈 and diagonal elements of𝜆, respectively. If we perform a QR decomposition of ¯𝑈^(ℓ), the top left 𝜌_ℓ × 𝜌_ℓ block is diagonal and the lower(𝑁 −𝜌_ℓ) rows are zero. Since ¯𝑈^(ℓ) is only an𝑁 ×𝜌_ℓ matrix, fewer Givens rotations are needed to perform the decomposition than in the full case. Specifically, it at worst requires ^𝑁₂

− ^{𝑁−𝜌ℓ}

2

rotations. Thus, as stated in the earlier discussion, the quantum circuit needed to implement rotation also contains that same number of gates in𝜌_(ℓ) layers.

In the quantum algorithm proposed in the main text, one rotates from the basis of one Cholesky vector to another. This rotation, explicitly given by𝑈^†(ℓ)𝑈^(ℓ+1), can be reduced into a single operation, corresponding to a 𝜌^(ℓ) × 𝜌^(ℓ+1) matrix. Using the same Givens rotation decomposition as above, we find that the costs are found by making the substitutions𝜌_(ℓ) → 𝜌_(ℓ+

1) and 𝑁 → 𝜌_ℓ.