PHYSICAL REVIEW LETTERS 127, 030503 (2021) Efficient Estimation of Pauli Observables by Derandomization Hsin-Yuan Huang ,1,2,* Richard Kueng,3 and John Preskill1,2,4,5 1Institute for Quantum Information and Matter, Caltech, Pasadena, California 91125, USA 2Department of Computing and Mathematical Sciences, Caltech, Pasadena, California 91125, USA 3Institute for Integrated Circuits, Johannes Kepler University Linz, A-4040, Austria 4Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, California 91125, USA 5AWS Center for Quantum Computing, Pasadena, California 91125, USA (Received 19 March 2021; accepted 14 June 2021; published 16 July 2021) We consider the problem of jointly estimating expectation values of many Pauli observables, a crucial subroutine in variational quantum algorithms. Starting with randomized measurements, we propose an efficient derandomization procedure that iteratively replaces random single-qubit measurements by fixed Pauli measurements; the resulting deterministic measurement procedure is guaranteed to perform at least as well as the randomized one. In particular, for estimating any L low-weight Pauli observables, a deterministic measurement on only of order logðLÞ copies of a quantum state suffices. In some cases, for example, when some of the Pauli observables have high weight, the derandomized procedure is substantially better than the randomized one. Specifically, numerical experiments highlight the advantages of our derandomized protocol over various previous methods for estimating the ground-state energies of small molecules. DOI: 10.1103/PhysRevLett.127.030503 Introduction.—Noisy intermediate-scale quantum (NISQ) randomized protocol that achieves our goal quite efficiently: devices are becoming available [1].Thoughlesspowerful For each of M copies of ρ, and for each of the n qubits, than fully error-corrected quantum computers, NISQ devices we chose uniformly at random to measure X, Y,orZ.Then used as coprocessors might have advantages over classical we can achieve the desired prediction accuracy with high computers for solving some problems of practical interest. success probability if M ¼ Oð3w log L=ε2Þ, assuming that For example, variational algorithms using NISQ hardware all L operators on our list have weight no larger than w have potential applications to chemistry, materials science, [11,12]. If the list contains high-weight operators, however, and optimization [2–10]. this randomized method is not likely to succeed unless M is In a typical NISQ variational algorithm, we need to very large. estimate expectation values for a specified set of operators In this Letter, we describe a deterministic protocol for fO1;O2; …;OLg in a quantum state ρ that can be prepared estimating Pauli-operator expectation values that always repeatedly using a programmable quantum system. To performs at least as well as the randomized protocol and obtain precise estimates, each operator must be measured performs much better in some cases. This deterministic many times, and finding a reasonably efficient procedure protocol is constructed by derandomizing the randomized for extracting the desired information is not easy in general. protocol. The key observation is that we can compute a In this Letter, we consider the special case where each Oj lower bound on the probability that randomized measure- is a Pauli operator; this case is of particular interest for ments on M copies successfully achieve the desired error ε near-term applications. for every one of our L target Pauli operators. Furthermore, Suppose we have quantum hardware that produces we can compute this lower bound even when the meas- multiple copies of the n-qubit state ρ. Furthermore, for urement protocol is partially deterministic and partially every copy, we can measure all the qubits independently, randomized; that is, when some of the measured single- choosing at our discretion to measure each qubit in the qubit Pauli operators are fixed, and others are still sampled X, Y,orZ basis. We are given a list of Ln-qubit Pauli uniformly from fX; Y; Zg. operators (each one a tensor product of n Pauli matrices), Hence, starting with the fully randomized protocol, we and our task is to estimate the expectation values of all L can proceed step by step to replace each randomized single- operators in the state ρ, with an additive error no larger than qubit measurement by a deterministic one, taking care in ε for each operator. We would like to perform this task each step to ensure that the new partially randomized using as few copies of ρ as possible. protocol, with one additional fixed measurement, has suc- If all L Pauli operators have relatively low weight (act cess probability at least as high as the preceding protocol. nontrivially on only a few qubits), there is a simple When all measurements have been fixed, we have a fully 0031-9007=21=127(3)=030503(7) 030503-1 © 2021 American Physical Society PHYSICAL REVIEW LETTERS 127, 030503 (2021) deterministic protocol. In numerical experiments, we find Lemma 1. (Confidence bound). Fix ε ∈ ð0; 1Þ (accu- that this deterministic protocol substantially outperforms racy) and 1 − δ ∈ ð0; 1Þ (confidence). Suppose that Pauli randomized protocols [12–16]. The improvement is espe- observables O ¼½o1; …; oL and Pauli measurements cially significant when the list of target observables includes P ¼½p1; …; pM are such that operators with relatively high weight. Further performance gains are possible by executing (at least) linear-depth circuits XL ε2 δ – O; P ≔ − h o ; P ≤ : before measurements [17 20]. Such procedures do, how- Confεð Þ exp 2 ð l Þ 2 ð3Þ ever, require deep quantum circuits. In contrast, our protocol l¼1 only requires single-qubit Pauli measurements, which are more amenable to execution on near-term devices. Then, the associated empirical averages (2) obey The manuscript is organized as follows. We first provide some statistical background, explain the randomized meas- jωˆ l − ωlðρÞj ≤ ε for all 1 ≤ l ≤ L ð4Þ urement protocol, then analyze the derandomization pro- cedure. We then provide numerical results showing how the with probability (at least) 1 − δ. derandomized protocol improves on previous methods. We See Supplemental Material Sec. B.1 for a detailed conclude with remarks and outlooks. Further examples and derivation [21]. We call the function defined in Eq. (3) details of proofs are in the Supplemental Material [21]. the “confidence bound.” It is a statistically sound summary Statistical background.—Let ρ be a fixed, but unknown, parameter that checks whether a set of Pauli measurements quantum state on n qubits. We want to accurately predict L (P) allows for confidently predicting a collection of Pauli expectation values observables (O) up to accuracy ε each. Randomized Pauli measurements.—Intuitively speaking, ωl ρ tr Oo ρ for 1 ≤ l ≤ L; 1 ð Þ¼ ð l Þ ð Þ a small confidence bound (3) implies a good Pauli estimation protocol. But how should we choose our M Pauli measure- O σ ⊗ ⊗ σ where each ol ¼ ol½1 ÁÁÁ ol½n is a tensor product ments (P) in order to achieve ConfεðO; PÞ ≤ δ=2?The of single-qubit Pauli matrices, i.e., ol ¼ [ol½1; …; ol½n] randomized measurement toolbox [12,13,16,22,23] provides with ol½k ∈ fI;X;Y;Zg. To extract meaningful informa- a perhaps surprising answer to this question. Let wðolÞ tion, we perform M (single-shot) Pauli measurements on denote the weight of Pauli observable ol, i.e., the number ρ 3n independent copies of . There are possible measure- Pof qubits on which the observable acts nontrivially: wðolÞ¼ n ment choices. Each of them is characterized by a full- k 1 1fol½k ≠ Ig. These weights capture the probability n ¼ weight Pauli string pm ∈ fX; Y; Zg and produces a of hitting ol with a completely random measurement string: n n q ∈ 1 wðolÞ random string of outcome signs m f g . Probp½ol▹p¼1=3 . In turn, a total of M randomly p 1 ≤ m ≤ M Not every Pauli measurement m ( )pro- selected Pauli measurements will, on average, achieve o vides actionable advice about every target observable l wðolÞ EP½hðol;PÞ¼M=3 hits, regardless of the actual Pauli (1 ≤ l ≤ L). The two must be compatible in the sense observable ol in question. This insight allows us to compute that the latter corresponds to a marginal of the former; expectation values of the confidence bound (3) i.e., it is possible to obtain ol from pm by replacing I some local nonidentity Pauli matrices with .Ifthisis L o ▹p p X the case, we write l m and say that measurement m wðolÞ M EP½ConfεðO; PÞ ¼ ð1 − ν=3 Þ ; ð5Þ “hits” target observable ol. For instance, X; I ; I;X ; ½ ½ l¼1 ½X; X▹½X; X,but½Z; I; ½I;Z; ½Z; Z ▹ ½X; X.Wecan approximate each ωlðρÞ by empirically averaging (appro- 2 where ν ¼ 1 − expð−ε =2Þ ∈ ð0; 1Þ. Each of the L terms is priately marginalized) measurement outcomes that exponentially suppressed in ε2M=3wðolÞ. Concrete realiza- belong to Pauli measurements that hit ol, tions of a randomized measurement protocol are extremely 1 X Y unlikely to deviate substantially from this expected behavior ωˆ q j ; (see, e.g., [11]). Combined with Lemma 1, this observation l ¼ h o ; p ; …; p m½ ð2Þ ð l ½ 1 MÞ m∶o ▹p l m j∶ol½j≠I implies a powerful error bound. Theorem 1. — P (Theorem 3 in Ref. [11].) Empirical h o ; p ;…;p M 1 o ▹p ∈ 0;1;…;M averages (2) obtained from M randomized Pauli measure- where ð l ½ 1 MÞ¼ m¼1 f l mg f g counts how many Pauli measurements hit target observ- ments allow for ε-accurately predicting L Pauli expectation O ρ ; …; O ρ ε able ol. values trð o1 Þ trð oL Þ up to additive error given wðolÞ 2 It is easy to check that each ωˆ l exactly reproduces ωlðρÞ that M ∝ logðLÞ maxl 3 =ε .