Derandomized shallow shadows: Efficient Pauli learning with bounded-depth circuits

TL;DR

The derandomized shallow shadows algorithm is presented, using shallow circuits to rotate into measurement bases, and its results indicate that in addition to being an efficient, low-depth, stand-alone algorithm, DSS can also benefit many larger quantum algorithms requiring estimation of multiple non-commuting observables.

Abstract

Efficiently estimating large numbers of non-commuting observables is an important subroutine of many quantum science tasks. We present the derandomized shallow shadows (DSS) algorithm for efficiently learning a large set of non-commuting observables, using shallow circuits to rotate into measurement bases. Exploiting tensor network techniques to ensure polynomial scaling of classical resources, our algorithm outputs a set of shallow measurement circuits that approximately minimizes the sample complexity of estimating a given set of Pauli strings. We numerically demonstrate systematic improvement, in comparison with state-of-the-art techniques, for energy estimation of quantum chemistry benchmarks and verification of quantum many-body systems, and we observe DSS's performance consistently improves as one allows deeper measurement circuits. These results indicate that in addition to being an efficient, low-depth, stand-alone algorithm, DSS can also benefit many larger quantum algorithms requiring estimation of multiple non-commuting observables.

Figures (9)

• Figure 1: Derandomized shallow shadows (DSS) algorithm. Given a measurement budget of $100$ measurements of depth $d=3$ and $30$ Pauli strings to learn, the classical DSS algorithm specifies the measurements we should make. (a) Derandomization of the $17$th measurement circuit: each measurement circuit is initially $d$ layers of random Clifford gates (circuit i). The algorithm goes gate-by-gate, first fixing each random two-qubit gate (circuit ii), then fixing each single-qubit gate (circuit iii), and ending with a fully deterministic circuit (circuit iv) where all single qubit rotations are decomposed into S and H gates nielsen_chuang_2010. (b) As this derandomization subroutine is performed on each of our $100$ measurement circuits, the variance of learning our $30$ Pauli strings decreases. The inset shows how the variance decreases with each gate that is fixed in the $17$th measurement. The greatest improvement comes when the final layer of gates is derandomized: suddenly we no longer have random gates and thus learn a fixed set of Paulis. (c) We show how many times, out of our 100 measurements, we probabilistically learn each of our 30 Paulis of interest (indices labeled with a star). As we derandomize our circuit, our probability of learning each starred Pauli peaks.
• Figure 2: Efficient ground state energy estimation under various quantum chemistry Hamiltonians. (a) With only one layer of two-qubit gates ($d=1$), DSS already outperforms previous state-of-the-art techniques for estimating the ground state energy of various molecules (parentheses indicate number of qubits). We plot the estimation error after 1000 measurements, and to avoid fluctuations due to non-trivial variance, the estimation errors reported are averaged over many simulations. We compare to locally-biased classical shadows (“LBCS”) hadfield2022measurements, Random Pauli derandomization (“Derand”) huang2021efficient, shallow shadows (“Shallow”) bertoni2022shallowippoliti2023operatorhu2023classical, and overlapped grouping measurement (“OGM”) wu2023overlapped. Notice that we consider estimating the ground state energy of $H_2$ with two different representations: 4 and 8 qubits. This corresponds to how one chooses different sets of molecular orbitals when discretizing real space. (b) Measurement circuits ($d =1$) for 100 measurements on $H_2$ (4 qubits). We find that groupings naturally emerge: our DSS algorithm groups Paulis which are simultaneously-diagonalizable under shallow circuits. For 100 measurements, DSS suggests taking $91$ measurements in the $Z$ basis (yellow) and $9$ in the bell basis (pink).
• Figure 3: Variational quantum simulation of the Hubbard model. We consider estimating the variance of the Hubbard Hamiltonian at various system sizes, which requires estimating the terms of $H^2$. In this figure we plot the number of measurements required to learn each term in $H^2$ at least 25 times. (a) Here, we consider the Hubbard model on 12 qubits. We normalize the number of experiments by the number required at $d=0$ (1236 experiments). Each point is therefore the percentage, of the number of measurements required at $d=0$, we need as we increase our measurement ansatz depth. As we increase the depth, we require fewer measurements, as expected, because we can simultaneously learn more of our Paulis in $H^2$. (b) We also fix the depth and modulate the system size. We find that at large system sizes our DSS protocol is most efficient -- in particular, DSS with only one layer of two-qubit gates ($d=1$) already outperforms all other strategies at 13 qubits. We expect that as we increase the depth of our ansatz, the required number of experiments will continue to decrease -- see blue gradient.
• Figure A.1: Bell basis example. Imagine wanting to learn the $n=8$ qubit Pauli strings $IIIXIIIX$, $IIIYIIIY$, and $IIIZIIIZ$ and allowing $N=100$ measurement circuits up to depth $d=4$. The DSS algorithm outputs $100$ copies of the measurement circuit, which rotates into the Bell basis on the fourth and final qubits. The bell basis simultaneously diagonalizes $XX$, $YY$, and $ZZ$, and therefore DSS finds the optimal outcome. This figure depicts what a single measurement circuit looks like as it is being derandomized. First the two qubit gates are fixed and then the single qubit gates are fixed. It is impressive that even though we allow depth $d=3$, DSS can find the optimal, lower-depth solution.
• Figure D.2: Example circuit and its associated tensor network. Consider the measurement ensemble $\mathcal{U}$ defined by this example circuit. Using tensor network techniques, we can solve for the probability $p(P)$ that this measurement ensemble measures the Pauli string $P = P^{(1)} \otimes P^{(2)}$. Looking at the circuit, we can immediately notice that the circuit diagram (left) maps to a network of tensors (right). Each gate, measurement, and input Pauli string becomes a tensor, as defined in the main text. We give examples of what some of the tensors look like on the right (see Eq. \ref{['eqn:TN_hadamard_tensor']}, Eq. \ref{['eqn:TN_twirl_single_qubit']}, and Eq. \ref{['eq:v_measurement']}). This setup allows us to directly evaluate $p(P)$ for this example ensemble.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Definition 1: DSS Initial Ansatz
• Definition 2: Pauli weight for measurement ensembles $\mathcal{U}_i$
• Definition 3: DSS Cost function
• Definition 4: Hitting Count
• Lemma 1: Confidence Bound
• proof
• Theorem 1: Performance guarantee from COST function
• proof