Measurement-Based Time Evolution for Quantum Simulation of Fermionic Systems

Quantum simulation using time evolution in phase estimation-based quantum algorithms can yield unbiased solutions of classically intractable models. However, long runtimes open such algorithms to decoherence. We show how measurement-based quantum simulation uses effective time evolution via measurement to allow runtime advantages over conventional circuit-based algorithms that use real-time evolution with quantum gates. We construct a hybrid algorithm to find energy eigenvalues in fermionic models using only measurements on graph states. We apply the algorithm to the Kitaev and Hubbard chains. Resource estimates show a runtime advantage if measurements can be performed faster than gates, and graph states compactification is fully used. In this letter, we set the stage to allow advances in measurement precision to improve quantum simulation.


I. INTRODUCTION
Unbiased quantum simulation [1,2] of intractable models aids in validating approximations. Compelling open problems include the two-dimensional Hubbard model of the cuprates and, more generally, materials and quantum chemistry models [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Such interacting fermionic models are typically NP-hard because they suffer from the fermion sign problem [18] and are generally parameterized as: where c † j creates a fermion in quantum state j (a composite index for spin, lattice site, etc.) and w ij (V ijkl ) is the single (two)-particle Hamiltonian matrix element. Since they are NP-hard, classical simulation time increases exponentially with particle number. Unbiased quantum simulation of models captured by H will therefore offer high impact benchmarks. Variational quantum algorithms offer promise on near-term devices [19] because they can be used to rigorously bound ground state energies.
Recent work [20] combines a variational quantum algorithm with measurement-based quantum computing (MBQC) [21,22] for efficient management of variational ansatz states. MBQC starts with a resource state, e.g., a graph state such as the square lattice cluster state (SLCS), formed by taking qubits aligned along the Pauli-x direction and then entangling them pairwise with controlled-Z gates. All quantum algorithms can then be executed using just single-qubit measurements on the resource state. MBQC-based variational algorithms [20] can therefore use measurements to bound ground state energies.
Phase-estimation-based quantum simulation algorithms [4,[23][24][25][26][27] can go beyond variational bounds to yield exact eigenfunctions and eigenvalues of H for use in benchmarking excited state properties. In circuit-based quantum computing (CBQC), such algorithms take an input wave function |ψ I and repeatedly apply quantum gates to time-evolve H with M small time steps δt g to eventually extract information. Quantum algorithms based on this procedure yield an advantage over classical algorithms but for runtimes that increase exponentially with the required bit precision in, e.g., eigenvalues. * Email address:scarola@vt.edu Long runtimes can be prohibitive [11,12,28] if, for N g gates per time step, the qubits cannot be kept coherent for long execution times, T C ∼ M N g δt g .
We propose revisiting phase-estimation-based quantum simulation runtime from the MBQC perspective. We consider the following regime: (i) A large number of qubits are available, (ii) the time taken for an accurate single-qubit measurement δt m can be made small enough to avoid decoherence of the resource state, and (iii) the entangling gates are performed in parallel mostly at the beginning. Assumption (iii) allows slow/error-prone entangling gates to be implemented and error corrected in a time that is negligible compared with the time to execute all measurements.
In this letter, we explicitly map real-time evolution in CBQC (repeated application of gates that take a finite amount of time) to repeated measurement in MBQC [22]. To this end, we make the following advances: (i) We construct a route to use MBQC to effectively time-evolve H using just singlequbit measurements. We show that long effective time evolution corresponds to M sequential measurement rounds in MBQC, thus requiring coherence among non-measured qubits for a total time, T M ∼ M N m δt m , where N m is the number of measurements per round. (ii) We construct an example hybrid MBQC algorithm with a quantum phase-estimationbased subroutine that yields exact eigenenergies: quantum eigenvalue estimation using an offline (classical) time series [6,29]; see Fig. 1(a). (iii) We apply the algorithm to solve the Kitaev [30,31] and Hubbard [32] chains because they can be solved exactly and can therefore be accurately checked as first implementations. To compare T M and T C for our algorithm, we compute scaling of MBQC measurement time and precision costs as well as gate counts in an equivalent CBQC algorithm.
Our central finding is that MBQC can yield a runtime advantage over CBQC, i.e., T M < T C , by shifting the burden of requiring low δt g but high-fidelity gates in CBQC simulation to the requirement of low δt m and high single-qubit measurement precision in MBQC simulation. Figure 1(b) summarizes our finding by showing that, if δt g /δt m is large, MBQC will have shorter runtimes. Here, N m /N g is set by the algorithm. We find that graph state compactification [34] can yield hybrid MBQC algorithms with N m /N g = 1. In this letter, therefore, we establishe a route to use improvements in quantum sensing [35] to advance the state of the art in quantum simulation with effective time evolution.

II. MEASUREMENT-BASED TIME EVOLUTION
Time evolution of Hamiltonians containing noncommuting terms H 1 and H 2 requires a decomposition. The firstorder Trotter-Suzuki decomposition is simplest [36,37]: Here the time step t/M is repeated M times until the output state is converged within a tolerance δ T [38], and = 1.
To map between fermions and qubits in H, we use the Jordan-Wigner (JW) transformation [39]: y ]/2, where σ a with a ∈ {x, y, z} are the Pauli matrices. Long JW strings containing N qubits can arise in certain models, e.g., those with long-range hopping/interaction in H. Longer-range terms allow simulation of higher-dimensional fermionic models H because they map to one-dimensional chains with long-range hopping and long-range interaction. Time evolution of a string requires the ability to execute nontrivial unitaries: R aj ], where θ is a rotation angle. The JW transformation enables construction of a time-toangle mapping for MBQC simulation. Figure 1(c) shows an example measurement pattern needed for time evolution of a hop between neighboring sites, c † 1 c 2 +c † 2 c 1 . In the absence of the central measurement (star), the measurement pattern swaps information on qubits 1 and 2 [22,33]. However, the additional adaptive measurement in the second round of measurements withφ 1,2 on the central qubit (star) incorporates results from the first round to yield [22] R (1,2) zz (θ)|ψ I , where θ defines the part not relying on random measurement outcomes inφ 1,2 . This operation is a time propagator, and one can show, see Supplemental Material [33], that, with a few more measurements, this measurement pattern effectively time-evolves a hop between sites 1 and 2. The number of measurements and qubits needed for effective time evolution on a SLCS, e.g., the left side of Fig. 1(c), can be significantly reduced. The Gottesman-Knill theorem [40] shows that all qubits with Pauli-x measurements can be excluded since Clifford operations can be efficiently executed classically. After mathematically removing local Pauli measurements, the SLCS maps to a compactified cluster state (CCS) [34]. The mappings show that a much smaller graph state is needed. For example, the right side of Fig. 1(c) shows an equivalent execution of R (1,2) zz (θ) (see Supplemental Material [33] for a proof), where the number of qubits (measurements) reduces from 12(10) to 5(3). In general, a CCS offers a reduction in measurement and qubit overhead for executing effective time evolution using R (1,2···N ) zz···z (θ) by as much as O(N 2 ), depending on which CCS is chosen. We construct example time-evolution subroutines on SLCSs with the understanding that use of a CCS reduces the number of required qubits and measurements at the expense of modifying qubit connectivities which is efficiently programmable [41].

III. KITAEV CHAIN
We construct an MBQC subroutine for time evolution of an example model with noncommuting terms, the Kitaev chain  Fig. 1(c). Adaptive measurements are carried out with the angles defined in Eq. (4). The indices j, k ∈ {1, 2, 3, 4} are assigned along the direction of information flow (red arrows). Measurement angles denoted by stars execute effective time evolution, while other shapes denote measurements to perform rotations at the ends of the Jordan-Wigner (JW) strings. All qubits but inputs with Pauli-x measurements (open boxes) can be removed in compactified cluster states (CCSs). (b) The same but for a two-site Hubbard chain with angles defined by Eq. (7) and j, k, l ∈ {1, 2, 3, 4}. [30,31]: where w ≥ 0 is the hopping and pairing energy, µ ≥ 0 is the chemical potential, and δn j = c † j c j − 1/2. The ground state exhibits a quantum phase transition at µ = 2w between a nontopological strong-coupling phase (µ > 2w) and a topological weak-coupling phase (µ < 2w).
We map fermions to qubits to construct both circuit-and measurement-based time propagators. The JW transformation maps H K to the quantum Ising model. The first-order Trotter- where g µ = µ/(2w) and is a measurement angle. Equation (2) can be implemented in two different ways: using real-time evolution in CBQC or effective time evolution in MBQC, where M dictates the circuit or measurement depth, respectively. Equation (3) is central because it maps real time t to measurement angle. We use the stabilizer formalism to map Eq. (2) to effective time evolution in MBQC. Figure 2(a) shows the measurement pattern implementing Eq. (2) to time-evolve input qubits [1][2][3][4] (open circles) with just single-qubit measurements. The measurement angles in the x-y plane arē where ψ r ∈ {±α, ±β, γ} for r = ±1, ±2, 3, −α = β = γ = π/2, and P θ = (−1) S K θ . Here, S K θ accumulates all measurement outcomes during single-qubit measurements and is derived in the Supplemental Material [33]. The measurement outcomes are also used for offline processing with a byproduct operator, see Supplemental Material [33], that defines the basis for interpreting output measurements.
The left, middle, and right panels depict measurements [stars in Fig. 2(a)] that entangle input qubits 1-2, 2-3, and 3-4, respectively. The measurement pattern in Fig. 2(a) and Eq. (4) define the full effective time-evolution algorithm for a Kitaev chain of any N or M because additional panels in Fig. 2
Both examples, Eqs. (4) for H K and Eqs. (7) for H H , demonstrate constraints on effective time evolution. Long effective time evolution from a larger number of Trotter steps in MBQC corresponds to smaller measurement angles since φ M ∝ 1/M . Repeated small-angle measurements (long effective time evolution) in MBQC therefore require improvements in qubit measurement precision as opposed to faster gates in CBQC.

V. EIGENVALUE ESTIMATION
To demonstrate resource requirements, we construct a minimal hybrid quantum eigenvalue estimation algorithm by combining MBQC subroutines with an offline time-series estimator [ Fig. 1(a)]. A |ψ I close to a desired eigenstate is fed into the MBQC time-evolving subroutine yielding ψ I |e −iHt |ψ I if the output qubits are measured using quantum state tomography (or an ancilla qubit [6]) to find the wavefunction phase relative to the input qubit basis. The MBQC output is obtained L times and used in a classical discrete Fourier transform: Re e (iωm−η)tn ψ I |e −iHtn |ψ I , where we define t n = nδt, ω m = mδω (n, m = 0, 1, · · · , L− 1) in units of δt and δω satisfying δωδt = 2π/L. Peaks in A(ω) yield eigenvalues of H to within δ T . We introduce the broadening parameter η > 0 for visualization of Lorentzian peaks and as a proxy for experimentally limited resolution.
The main panel in Fig. 3 shows a demonstration result from a simulation using H K in Eq. (8), where several eigenvalues are returned as peaks. One can show, see Supplemental Material [33], that peak centers are intact while peak weights are shifted for certain types of measurement errors. Figure 3 uses large L and M for clarity but in practice, L and M can be lowered. They are minimized by restricting the search to just the ground state energy, while three independent algorithm input parameters δω, L, and M must be chosen to meet three tolerances: (i) δω should be smaller than η to resolve peak structure, (ii) A sum rule tolerance δ F > 1 − δω sets L, and (iii) M is set by requiring δ T to be much smaller than the first spectral gap.

VI. MEASUREMENT PRECISION
The number of Trotter steps yields the measurement depth and sets φ M . Large M improves Trotter accuracy at the expense of requiring improved measurement precision (small φ M ). To estimate the minimum M needed to obtain ground state energies, we consider H K with g µ = 0.01 − 0.4. We find empirically, see Supplemental Material [33], that, for each n in Eq. (8), the minimum M varies from 1.8 × 10 3 (g µ = 0.01) to 7.8 × 10 4 (g µ = 0.4) to resolve the ground state energy of H K to within 1% of the spectral gap (δ T = 10 −2 ) for η = 0.02w, δω = 0.01w, L = 46, and N = 4. We have checked N ≤ 8 with other η, δω, and L combinations and obtained similar results for M . In general, the M needed will depend on the model, model parameters, tolerances, and scales as O((N t n ) 2 δ −1 T ) [43], thus implying that the required measurement depth and precision to execute effective MBQC time evolution can become demanding [44].
Given bounds on M , we can estimate measurement precision requirements for H K . Here, φ M depends on n. The , in three scenarios(rows): (i) MBQC on an SLCS including all Pauli-x and adaptive measurements, (ii) MBQC on a CCS with the least number of measurements, and (iii) CBQC. In (i) and (ii), measurements on input/output qubits are not counted. (ii) and (iii) show the same scaling (Nm/Ng = 1) for two different experimental processes, measurements and two-qubit gates. Approach We empirically find, see Supplemental Material [33] (far from the critical point at g µ = 1), χ n 0.14, thus allowing the use of Eq. (3). The smallest measurement angle increment needed in Eq. (4) is g µ χ n . We find g µ χ n 4.8 × 10 −4 for all g µ < 1 and n. We therefore see that a large M requires small angle measurements as we implement effective time evolution.

VII. MEASUREMENT AND QUBIT OVERHEAD
The measurement subroutines defined by Eqs. (4) and (7) allow estimates of resource requirements in our hybrid quantum eigenvalue estimation algorithm. Table I shows, see Supplemental Material [33], that, for the local models considered here, a CCS will have N m /N g = 1. However, with nonlocal qubit terms, e.g., for nonlocal hopping in H, MBQC with a CCS will have an O(N ) advantage in measurement vs gate counts in CBQC unless nonlocal gates are used to implement the JW strings [45]. The number of qubits needed is O(M ) larger for MBQC than for CBQC. MBQC qubit overhead can be lowered by re-entangling measured qubits [21].

VIII. DISCUSSION
Our demonstration algorithms show that unbiased quantum simulation using effective time evolution is possible using only single-qubit measurements on graph states. We find that long MBQC effective time evolution for use in quantum simulation requires high measurement precision to be useful in benchmarking approximate classical algorithms. Alternative time-evolution decompositions [16,[46][47][48][49] will lower overhead.
The above algorithms have a low error threshold [56,57]. An improvement with higher thresholds is available [58,59]. The above algorithms can also be used in conjunction with an adaptive Bayesian algorithm (instead of a time series) in eigenvalue estimation learning certain types of error [60,61].
Finally, applications to higher-dimensional fermionic models are highly desired. Nearest neighbor hoppings/interactions in a higher-dimensional fermionic lattice can be mapped to long-range hoppings/interactions in a chain [6]. After mapping, our hybrid MBQC algorithm can be applied to the chain at the expense of increasing the length of JW strings. In this section, we prove that, as stated in the main text, the measurement pattern in the left side of Figure 1(b) effectively time-evolves a Pauli-z string between sites 1 and 2. Pauli-z strings are needed for Jordan-Wigner (JW) strings. We also prove that Figure 1(b) generalizes to time evolution of longer-range strings on a larger square lattice cluster (SLCS) state using only O(1) adaptive measurements.
We start with the definition of a SLCS. Consider a connected subset of a square lattice L 2 with vertices V(L 2 ) and edges E(L 2 ). The SLCS is defined by [1]: where CZ . The SLCS satisfies the eigenvalue equation K j |G SLCS = |G SLCS with the stabilizer given by Here the Pauli matrices are abbreviated by (X, Y, Z) ≡ (σ x , σ y , σ z ), and N j indicates the nearest neighbors of site j. An SLCS can serve as a platform to implement unitary gates equivalent to circuit-based quantum computing (CBQC) gates. The SLCS is composed of three sections: an input, a body, and an output. The number of input qubits is the same as the number of qubits defining the qubit Hamiltonian. The number of output qubits is the same as the number of input qubits. A central idea in measurement-based quantum computing (MBQC) is that a sequence of single-qubit measurements on the input and body sections trigger information flow from the input to the output [2]. The measurement pattern in the body determines which gate operation is encoded. Qubits in the input and body sections may be measured in an adaptive/non-adaptive basis in the x-y plane.
After measurements we must define the measurement basis using a byproduct operator that returns equivalent CBQC gates. We consider the projected SLCS: |G proj = j P (j) sj (φ j )|G SLCS , where j spans all sites in the input and body sections, and the projector is defined by P (1, 1), and X = Xx + Yŷ + Zẑ. In Ref. [3], it was shown that |G proj is governed by a set of eigenvalue equations: where j (k) is an index for the input (output) qubits, spanning from 1 to N . Once we find U g , λ x,j , λ y,j in Eqs. (S3) and (S4), the output wavefunction has the connection to the input: |ψ O = U g U Σ |ψ I up to the U (1) phase factor. The byproduct operator is defined by which adjusts the output qubit basis. U g and U Σ can be switched using Pauli propagation relations, e.g., XZ = −ZX and Y Z = −ZY . Then, the output wavefunction can be refined by |ψ O = U Σ |ψ O = U g |ψ I to retain the right basis. If we match U g as in CBQC, we can derive the connection between the measurement angles and the model parameters.
We now apply the above formalism to the construction of JW strings. As mentioned in the main text, time evolution of an N -site JW string requires the ability to execute N -qubit rotation gates: where θ is a rotation angle and a j ∈ {x, y, z}. It turns out that Eq. (S6) has the decomposition: where U x (α j ) executes the Euler rotation of a qubit at site j to adjust the qubit basis. In the following two subsections, we use the stabilizer formalism to derive the mathematical details of the measurement patterns for R (1,2) zz (θ) and R (1,2,3) zzz (θ) which play a central role in MBQC simulation of the Kitaev and Hubbard chains (A proof for N = 2 is outlined in Ref. [3], but we include this case here for consistency).
The full JW string takes R (1,2···N ) zz···z (θ) and then applies single-qubit rotations to the end qubits (1 and N ). These rotations can also be implemented with measurements. To implement single-qubit rotation gates with Euler angles (α, β, γ), we use [3]: , with the measurement angles given bȳ where the index j inψ j and s j is defined along the 5-qubit cluster chain with the input (output) qubit at j = 1(5). In the following two sections, we construct the measurement patterns needed to implement R with the understanding that we must follow up with single-qubit rotations to the end qubits as shown in Sec. III.
This concludes the construction of the measurement pattern needed to implement R (1,2,3) zzz (θ). By comparing the equations forφ 1,2,3 andφ 1,2 , we therefore see how to systematically increase the length of the string with larger cluster states. Figure S1 also shows that only one adaptive measurement is needed to implement long strings because we can increase N inductively.

II. COMPACTIFICATION OF SQUARE LATTICE CLUSTER STATE
In this section, we use the theorem on local Pauli measurement to prove that the two-qubit rotation operation implemented with the measurements on the left side of Figure 1(b) in the main text is equivalent to the operation implemented by the measurements depicted on the right side. We start by referring to the Gottesman-Knill theorem [4] showing that Clifford operations can be efficiently executed classically. This suggests carrying out the first round of Pauli-x projectors in the body section of the SLCS in advance to map the original SLCS to the compactified form with simpler connectivity. This process is formulated in the following theorem [5]: A local Pauli projector on the qubit at site j in a SLCS yields a compactified cluster state (CCS) |G CCS on the remaining qubits: for a j ∈ {x, y, z} and m j = ±. Here, the compactified graph G CCS is defined by for arbitrary choice of k ∈ N j (nearest neighbors to j), up to the local unitaries: which are composed of the Clifford and non-Clifford parts. Here, √ ±iσ a ≡ e ±i(π/4)σa for a ∈ {x, y, z}.
In the above theorem, G CCS is involved in the local complementation (LC) for the case of a j = x, y. The LC of G SLCS at a site j, i.e., τ j : G SLCS → τ j (G SLCS ), is obtained by complementing the subgraph of G SLCS induced by N j (i.e., disconnecting/connecting qubits belonging to N j if they are originally connected/disconnected) and leaving the rest of parts unchanged. The corresponding LC-equivalent SLCS is defined by |τ j (G SLCS ) = U  aj ,m , because local unitaries always intervene. We note that Pauli propagation is governed by a different rule for the Clifford and non-Clifford gates. For the Clifford gates, Pauli propagation at most flips the sign of projector outcomes: P x,± Z = ZP x,∓ , P y,± Z = ZP y,∓ . On the other hand, for the non-Clifford gates, it makes a more drastic impact, i.e., the reorientation of directions: We now revisit the measurement pattern for the 2-qubit rotation gate [see Figure S1(a) or the left side of Figure 1(b) in the main text]. To find the CCS [shown in the right side of Figure 1(b) in the main text], we use the above theorem to project qubits at sites j 1 = (2, 1), j 2 = (2, 3), j 3 = (3, 1), j 4 = (3, 3), j 5 = (4, 1), j 6 = (4, 3), j 7 = (3, 2). We take the following steps as summarized in Figure S2: Step 1: Pauli-x projector is applied at site j 1 = (2, 1) (red box) with special neighbor k 1 = (3, 1) (blue box) to find x,m1 . (1a) LC of G SLCS at k 1 , (1b) LC of (1a) at j 1 , (1c) exclusion of j 1 from (1b), and (1d) LC of (1c) at k 1 .
Step 3: A Pauli-x projector is applied at site j 3 = (3, 1) in G (2) CCS . Since P (j3) x,m3 does not commute with m 1 iY k1(=j3) (obtained from Step 1), Pauli propagation is used to obtain: P (j3) x,m3 CCS . Thus Pauli-z measurement is effectively carried out at j 3 , leading to the exclusion of j 3 .
Step 4: A Pauli-x projector is applied at site j 4 = (3, 3) in G CCS . Since P (j4) x,m4 does not commute with m 2 iY k2(=j4) (obtained from Step 2), Pauli propagation is used to obtain: P CCS . Thus Pauli-z measurement is effectively carried out at j 4 , leading to the exclusion of j 4 .
We note that the final result G CCS depends on the choice of special neighbors (k 1 , k 2 , k 5 , k 6 , k 7 ). But it turns out that any variation belongs to the same LC-equivalent class up to local Clifford unitaries. This completes the constructive proof showing the equivalence of the results of measurements depicted on left and right sides of Figure 1(b) in the main text.

III. MBQC SUBROUTINES FOR FERMION SYSTEMS
In this section, we derive two expressions stated in the main text: (1) the signs for the measurement angles, S K θ and S H θ , and (2) the byproduct operators of the MBQC subroutines for the Kitaev and Hubbard chains. We also prove that, as stated in the main text, the measurement pattern for the Kitaev and Hubbard chains can be concatenated to time-evolve larger N or M .
Other exponents for the single-qubit rotation gates are defined in the hierarchical form [see Figure S3 Here, input and output qubits are indexed by (j1, j2) = (j, j + 1) for a given j in (b), and (N1, N2) = (N − 1, N ) for a given N in (c).
We now concatenate the N = 2 result to N ≥ 3. We proceed in two steps: First, we build three types of measurement patterns for W Figure S4). Second, we combine them in a specific order for a given N . The resulting giant measurement pattern has a cascade structure flowing from the left top to the right bottom. It turns out that there is no change in Eq. (S38) but with generalization:φ The exponents in Eqs. (S39)-(S43) are modified into: and (11,3) , respectively. Here the superscripts in s (j) and S (j) are to be consistent with W , and δ j,j is the Kronecker delta. The exponents in the byproduct operator have forms similar to Eq. (S45): To complete the concatenation process, we need to reapply Pauli propagation to push all byproduct operators to the left side of all other unitary gates in the giant measurement pattern. In this process, the exponents in V (j) g are adjusted to accumulate further the measurement outcomes in V (j−1) g . The same thing also happens for the pairs: Consequently, some exponents are modified in the following way: This concludes our derivation of expressions discussed in the main text: (1) the signs for the measurement angles, S K θ , and (2) the byproduct operators for the Kitaev chain. We have also proven the statement that the measurement pattern for the Kitaev chain shown in the main text can be concatenated for larger chains.

B. Hubbard chain
We start with the first-order Trotterized form of e −iHHt [Eq. (6) in the main text]: It is convenient to recast Eq. (S54) into the MBQC-adaptive form by decomposing the three-qubit rotation gates into in conjunction with the Euler decomposition R y The number of single-qubit rotation gates can be reduced by applying Pauli propagation to the array of gates: . After some algebra, Eq. (S54) is rearranged into a form for use in concatenation of longer Hubbard chains: where we define two types of composite gates: . Coordinate systems for assigning qubit positions in the measurement pattern for the Hubbard chain, which can be built by combining two types of measurement patterns for (a) V Here, input and output qubits are indexed by (a) (j1, j2, j3, j4) = (2j − 1, 2j, 2j + 1, 2j + 2) for a given j, (b) (j1, j2, j3, j4) = (2N − 3, 2N − 2, 2N − 1, 2N ) for a given N . Figure S5(a)], it turns out that the measurement angles have the form: where ψ r ∈ {±α, ±β, ±γ, ±λ} for r = ±1, ±2, ±3, ±4, and P θ = (−1) S θ . The exponents for the three-qubit rotation gates are defined by Other exponents for the single-qubit rotation gates are defined in the hierarchical form:     (40,7) .

IV. ESTIMATION OF TROTTER STEPS AND MEASUREMENT PRECISION
In this section, we explicitly show the empirical calculation used to obtain the minimum number of Trotter steps, M , and the normalized measurement angle χ n ≡ nw/(δωLM ) discussed in the main text. Figure S7(a) plots the minimum value of M needed to meet tolerances for the Kitaev chain for several different values of the chemical potential. We see that the M needed increases nearly linearly with the time step index. Figure S7(b) plots the corresponding measurement angles needed as a function of time step for several different chemical potentials. Here we see that, as stated in the main text, the largest measurement angle is below 2π (as needed), and that the smallest angle needed to be measured can become very small as g µ → 1. These graphs show how the bounds on M and χ n stated in the main text were obtained.

V. ESTIMATION OF RESOURCE REQUIREMENTS
In this section, we prove how we obtained the resources requirements for a single time step of Eq. (8) in the main text and for fixed M and N , shown in Table I in the main text. • CCS measurements: On a CCS, all Pauli-x measurements in the body section of SLCS are excluded. As before, we don't count measurements on input qubits lying at the end of information flow. For N = 2, the measurement count in Fig. S3(a) is reduced to 13. For N = 3, the measurement counts in Figs. S4(a) and (c) are reduced to 10 and 11, respectively. Merging Figs. S4(a) and (c), and further excluding Pauli-x measurement at input qubit 2 in Fig. S4(c), the total measurement count is reduced to 10 + 11 − 1 = 20. For N ≥ 4, the measurement count for Fig. S4(b) is reduced to 8. Merging Figs. S4(a)-(c) in the same way as before, and further excluding Pauli-x measurements at input qubits j (2 ≤ j ≤ N − 1) in Figs. S4(b) and (c), the total measurement count is reduced to 10 + 8(N − 3) + 11 − (N − 2) = 7N − 1. M -times repetition produces (7N − 1)M .
• Circuit-based gates: Circuit-based gates are counted by using Eqs. (S32)-(S36). It turns out that the result is consistent with the CCS measurement count.

B. Hubbard chain
• SLCS measurements: On an SLCS for a Hubbard chain (Fig. S5), all Pauli-x (open boxes) and adaptive measurements (boxes including symbols) are counted. For our purpose, we don't count measurements on input qubits lying at the end of information flow. For N = 2, we consider only Fig. S5(b). The measurement count is 168. For N ≥ 3, the N − 2 copies of Fig. S5(a), indexed by j (1 ≤ j ≤ N − 2), respectively, are stacked side by side, and we end up with Fig. S5(b) at the rightmost side. The measurement count for Fig. S5(a) is 154 for j = 1 or 156 for 2 ≤ j ≤ N − 2. The measurement count for Fig. S5(b) is 170. The total measurement count is 154 + 156(N − 3) + 170 = 156N − 144. M -times repetition produces (156N − 144)M .
• CCS measurements: On a CCS, all Pauli-x measurements in the body section of SLCS are excluded. As before, we don't count measurements on input qubits lying at the end of information flow. For N = 2, the measurement count in Fig. S5(b) is reduced to 36. For N ≥ 3, the measurement count for Fig. S5(a) is reduced to 34 for j = 1 or 36 for 2 ≤ j ≤ N − 2. The measurement count for Fig. S5(b) is reduced to 38. Merging Figs. S5(a) and (b) in the same way as before, and further excluding Pauli-x measurements on input qubits not lying at the end of information flow, the total measurement count is reduced to 34 + 36(N − 3) + 38 − 2(N − 2) = 34N − 32. M -times repetition produces (34N − 32)M .
• Circuit-based gates: Circuit-based gates are counted by using Eqs. (S57)-(S59). It turns out that the result is consistent with the CCS measurement count.