Hamiltonian Simulation by Qubitization

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Hamiltonian Simulation by Qubitization

Guang Hao Low

1

_{and Isaac L. Chuang}

2

1_{Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA}

2_{Department of Electrical Engineering and Computer Science, Department of Physics, Research Laboratory of Electronics,} Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

July 12, 2019

We present the problem of approximating the time-evolution operatore−iHtˆ to error, where the HamiltonianHˆ = (hG|⊗Iˆ) ˆU(|Gi⊗Iˆ)is the projection of a unitary oracleUˆ onto the state |Gi created by another unitary oracle. Our algorithm solves this with a query complexityO t+ log(1/)to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are d-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where Hˆ is a density matrix. A key technical result is ‘qubitization’, which uses the controlled version of these oracles to embed anyHˆ in an invariant SU(2)subspace. A large class of operator functions ofHˆ can then be computed with optimal query complexity, of whiche−iHtˆ is a special case.

1 Introduction

Quantum computers were originally envisioned as machines for efficiently simulating quantum Hamilto-nian dynamics. As HamiltoHamilto-nian simulation isBQP-complete, the problem is believed to be intractable by classical computers, and remains a strong primary motivation. The first explicit quantum algorithms for Hamiltonian simulation were discovered by Lloyd [1] for local interactions, and then generalized by Aharonov and Ta-Shma [2] to sparse Hamiltonians. Celebrated achievements over the years [3–7] have each ignited a flurry of activity in diverse applications from quantum algorithms [8–11] to quantum chemistry [12–18]. In this dawning era of the small quantum computer [19, 20], the relevance and necessity of space and gate efficient procedures for practical Hamiltonian simulation has intensified.

The cost of simulating the time-evolution operatore−iHtˆ _{depends on several factors: the number}

of system qubits n, evolution time t, target error , and how information on the Hamiltonian Hˆ is accessed by the quantum computer. This field has progressed rapidly following groundbreaking work in the fractional query model [21], which was the first to achieve query complexities that depend logarithmically on error. This was generalized by Berry, Childs, Cleve, Kothari, and Somma (BC-CKS) [22] to the common case where Hˆ = Pd

j=1αjUˆj is a linear combination of d unitaries1 with

an algorithm usingO log (d) log (τ /)_{log log (τ /)}

ancilla qubits and onlyO _{log log (τ /)}τlog (τ /)

queries, whereτ ← kα~k1t

andk~αk1=Pd_j=1|αj|is the usual L1norm. Subsequently, an extension was made tod-sparse

Hamil-tonians [6] that have at most d non-zero elements per row, also using O _{log log (τ /)}τlog (τ /)

queries with τ←dtkHkmax. A prominent open question in all these works was whether the additive lower bound Ω τ+_{log log (1/)}log(1/) 2 _{was achievable for any of these models.}

Recently, we found for thed-sparse model [7], which is popular in algorithm design by quantum walks [23], a procedure achieving the optimal trade-off between all parameters up to constant factors, with query complexityO τ+log (1/)3_{. This strictly linear-time performance with additive complexity} is a quadratic improvement over prior art for precision simulations whent= Θ(log (1/)). Moreover, the number of ancilla qubits required is independent ofτ and, which is another important practical improvement. The approach, based on Childs’ [4, 24, 25] extension of Szegedy’s quantum walk [26], required two quantum oracles: one accepting a rowj and columnkindex to return the value of entry

ˆ

Hjk to mbits of precision, and another accepting aj row and l sparsity index to return in-place the lth _{non-zero entry in row}_j_.

Unfortunately, the d-sparse model is less appealing in practical implementations for several rea-sons. First, it is exponentially slower than BCCKS when theUˆjare of high weight with sparsityO(2n).

Second, its black-box oracles can be challenging to realize. Avoiding the exponential blowup by ex-ploiting sparsity requires that positions of non-zero elements are both few in number and efficiently row-computable, which is not always the case. Third, the Childs quantum walk requires a doubling of thensystem qubits, which is not required by BCCKS. It is unclear whether our methodology could be applied to other formulations of Hamiltonian simulation, in contrast to alternatives that seem more flexible [27].

Ideally, the best features of these two algorithms could be combined, such as in Table 1. For example, given the decomposition Hˆ = Pd

j=1αjUˆj, one would like the optimal additive

complex-ity of sparse Hamiltonian simulation, but with the BCCKS oracles that are more straightforward to implement. Furthermore, one could wish for a constant ancilla overhead, of saydlog₂(d)e+ 2,

supe-1_{A particularly interesting and general class of systems are} _k_{-local Hamiltonians with} _m _terms. _{These are also} m2k_{-sparse, and can be expressed as a linear combination of}_≤_m₄k_{Pauli operators, each of which are unitary.}

2_{A slightly tighter lower bound is Ω(}_q_{) queries, where}_q_{= min}_{_q_∈_Z+_:_<_|_sin(_|_τ_|_/q₎_|q₌_O₍₍_{τ /q}₎q₎_}_[₆_,₂₁_{]. Note}

thatq∈ O(τ+ log (1/))∩Ω τ+_{log log (1}log(1/_/)₎.

3_{In the respective limits}_τ_{= Θ(1) and}_{= Θ(1), the query complexity is}_O log(1/) log log (1/)

andO(τ). Importantly, this

does not imply a complexity ofO τ+_{log log (1}log(1/_/)₎. The precise form of the upper bound matches the lower bound with

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Algorithm Model Ancilla qubits Query Complexity Gates per QueryO(·)

O(·) Oracle Primitive

LC [7] Sparse n+O(m) dt+_{log log (1/)}log (1/) Varies n+ poly(m)

BCCKS [22] P

jαjUˆj O(

log (d) log (k~αk1t/)

log log (kα~k1t/) )

kα~k1tlog (k~αk1t/)

log log (kα~k1t/) dC log (d)

LMR [5] Mixedρ n+ 1 t2/ Varies log (n)

Theorem 1 hG|Uˆ|Gi dlog2(d)e+ 2 t+

log (1/)

log log (1/) Varies log (d) Corollary 16 P

jαjUˆj dlog2(d)e+ 2 k~αk1t+

log (1/)

log log (1/) dC log (d) Corollary 17 Purifiedρ n+dlog₂(d)e+ 2 t+_{log log (1/)}log (1/) Varies log (n)

Table 1: Comparison of state-of-art with our new approaches (bottom three lines) for approximating e−iHtˆ _of

ˆ

H ∈ C2n×2n with error. The d-sparse simulation oracle describes entries ofHˆ with maximum absolute value

k_Hˆ_k_max _{= 1}_to_m _{bits of precision. The BCCKS oracle provides the decomposition} _Hˆ ₌ Pd

j=1αjUjˆ , and each ˆ

Uj is given a cost O(C). The LMR query complexity refers to samples of the density matrixρˆ= ˆH. This work generalizes the above with oraclesG|ˆ0i=|Gi ∈Cd_,_U_ˆ _∈_C2nd×2nd _{such that}_hG|_U_ˆ_|Gi_{= ˆ}_H_{, where}_k_{Hk ≤}_ˆ ₁_{. A}

rior to either algorithm. These improvements would greatly enhance the potential of early practical applications of quantum computation.

We achieve precisely this optimistic fusion via an extremely general procedure, made possible by what we call ‘qubitization’, that subsumes both and motivates new formulations of Hamiltonian simulation, as captured by this main theorem:

Theorem 1(Optimal Hamiltonian simulation by Qubitization). Let (hG|a⊗Iˆs) ˆU(|Gia⊗Iˆs) = ˆH∈

CN×N _{be Hermitian for some unitary}_U_ˆ _∈_CN d×N d_{and some state preparation unitary}_G_ˆ_|₀_i

a =|Gia∈

Cd_{. Then} _e−iHtˆ _{can be simulated for time} _t_{, error} _{in spectral norm, and failure probability} _O₍₎_,

using at most dlog₂(N)e+dlog₂(d)e+ 2 qubits in total, Θ(Q) queries to controlled-Gˆ, controlled-Uˆ, and their inverses, andO(Qlog (d))additional two-qubit quantum gates where

Q= min

q∈Z+ :≥ 4|t| q

q!2q =O

eτ

2q q

=O(t+ log (1/)). (1)

The optimality of the procedure follows by using oracles Gˆ and Uˆ that implement the Childs quantum walk. Furthermore, the transparent Hamiltonian input model of Theorem 1 significantly expedites the development of new useful formulations of Hamiltonian simulation. For instance, we easily obtain a new result for the scenario whereHˆ is a density matrixρˆ. Whereasρˆcan be produced by discarding the ancilla of some output from a quantum circuitGˆ, we instead keep this ancilla, leading to an unconditional quadratic improvement in time scaling, and an exponential improvement in error scaling over the sample-based Lloyd, Mohseni, and Rebentrost (LMR) model [5, 28], as summarized in Table 1. Though certainly a stronger model than was previously considered, the inputs to many quantum machine learning as well as quantum semidefinite programming [11] applications are actually of this more restricted form, and are thus enhanced.

In fact, Hamiltonian simulation is just one application of our main innovation: an approach we call the ‘quantum signal processor’, where the equation(hG|a⊗Iˆs) ˆU(|Gia⊗Iˆs) = ˆH inTheorem 1is

interpreted as a non-unitarysignal operatorHˆ encoded in a subspace of an oracleUˆ flagged by|Gia⊗

ˆ

Is. This general encoding defined in Definition 14 unveils a systematic approach toward exponential

quantum speedups with generic inputs, and is central to our results:

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Problem BCCKS [6] d-sparse [7] Evolution byρ QPE QLSP [9] Gibbs [10] ˆ

H Hamiltonian Hamiltonian Density matrix Unitary Matrix Hamiltonian ˆ

U Selects ˆUj Isometry ˆT SWAP Any Any Any

ˆ

G Uˆj coefficients Identity Purifiedρ Any Any Any

Solution e−iHtˆ _e−iHtˆ _e−iρtˆ _Decision _Hˆ−1 _e−βHˆ

Table 2: List of six example problems (top row), solvable using the quantum signal processor approach to compute an operator functionf[·]of HermitianHˆ =hG|_Uˆ_|Gi_{. Through qubitization, the scope of inputs to the Quantum}

Linear Systems Problem (QLSP) and Gibbs Sampling (Gibbs) can be anyHˆ of this form, either indirectly through Hamiltonian simulation, or directly through quantum signal processing. Quantum Phase Estimation (QPE) here decides whether eigenphasesθof an implemented unitary satisfy some property e.g. f(θ)≥1/2.

Definition 1 (Standard-form). A signal operator Hˆ with spectral norm k_Hˆ_{k ≤} ₁ _{is encoded in the}

standard-form if we may query a unitary oracleUˆ :Ha⊗Hs→ Ha⊗Hsand a unitary state preparation

oracle Gˆ|0ia = |Gia ∈ Ha with the property (hG|a⊗Iˆs) ˆU(|Gia ⊗Iˆs) = ˆH. We also assume query

access to the inverses and controlled versions ofU ,ˆ Gˆ.

The inputs to many problems, highlighted inTable 2, can be of this form. We introduce ‘qubitiza-tion’: the essential first step that converts this description ofHˆ into unitary evolution with properties that depend directly on Hˆ. When Hˆ is Hermitian, The oracles Gˆ and Uˆ are queried to obtain a Grover-like search parallelized over all eigenvaluesλofHˆ through theiterate, which is isomorphic to e−iYˆ⊗cos−1_{[ ˆ}_H]

in some subspace. Similar structures are of foundational importance to many quantum algorithms – the gap ∆ of eigenvalues λ = 1−∆ of Hˆ is amplified to cos−1₍_λ_{) =} _O₍√_∆) _{in the}

phase of the iterate, which resembles spectral gap amplification [30], the quantization of stochastic matrices [31], as well as Szegedy’s [26] and Childs’ [24] quantum walk. The key difference lies in the generalized encoding of the signal operator through anyGˆ andUˆ ofDefinition 1, instead of the more restrictive sparse Hamiltonian formulation.

Unlike Grover search, we do not seek to prepare some target state. Rather, we exploit a direct sum of Grover-like rotations that are each isomorphic to SU(2), to engineer arbitrary target functionsf(λ) of its overlapλ. The quantum signal processor exploits this structure to attack the often-considered problem of designing a quantum circuit Qˆ that queries Gˆ and Uˆ such that in the standard-form, (hG|a⊗Iˆs) ˆQ(|Gia⊗Iˆs) =f[ ˆH] for some target operator f[·]. Though this is accomplished in prior

art using the linear-combination-of-unitaries algorithm [25], that approach requires a case-by-case detailed analysis off to obtain theL1norm of its coefficients in a Taylor expansion, and has a success

probability that decays with the inverse square of this norm.

Our quantum signal processor computesf[ ˆH]with no such restrictions and with an optimal query complexity that exactly matches polynomial lower bounds for a large class of functions. We call this ‘quantum signal processing’, which generalizes our previous results [7] for d-sparse oracles to the standard-form and a larger class of functions. Thus generic improvements to all applications inTable 2can be expected in query complexity, ancilla overhead, and scope of possible signal inputs. In particular,Theorem 1follows directly from the query complexity of the choicef(λ) =e−iλt_{, which}

corresponds to applying−tsin (·)on eigenphases of the iterate.

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2 Overview of the Quantum Signal Processor

Since coherent quantum computation is restricted to unitary operations, one commonly finds a situ-ation such as inTable 2, where post-selection is required to accomplish some desired quantum state transformation. Consider some arbitrary input system quantum state|ψis∈ Hsand suppose that it is

transformed by some desiredsignal operator Hˆ like|ψis→Hˆ|ψis. In order to realize this operation,

which is non-unitary in general, it is necessary to embedHˆ in a larger Hilbert space.

This is embedding accomplished through some given unitarysignal oracleUˆ :Ha⊗ Hs→ Ha⊗ Hs

that acts jointly on Hs and another Hilbert space Ha. The signal oracle encodes Hˆ = (hG|a ⊗

ˆ

Is) ˆU(|Gia⊗Iˆs)in a subspace flagged by an ancillasignal state |Gia∈ Ha. This is the standard-form

ofDefinition 1, which appliesHˆ as follows.

ˆ

U|Gia|ψis=|GiaHˆ|ψis+

q

1− kHˆ|ψik2_|_G⊥

ψias, Uˆ =

_ˆ

H ·

· ·

, (hG|a⊗Iˆs)|G⊥ψias= 0. (2)

The signal state defines the measurement basis of the appended registerHa, which naturally divides

ˆ

U into two subspaces. First, HG = |Gi ⊗ Hs where the measurement succeeds with probability kHˆ|ψik2 _and_Uˆ_|_G_i_a_|_ψ_i_s _{is projected onto} |GiaHˆ|ψis

kHˆ|ψik . Second, the orthogonal complementHG⊥ where

the measurement fails. As probabilities are bounded by1, the signal operator Hˆ must have spectral norm kHˆk ≤1. Whenever the context is clear, we drop the ancilla and system subscripts, and use

|Gia⊗Iˆsand|Giinterchangeably. We representUˆ such that the top-left block is preciselyHˆ and acts

on an input state|Gψi ≡ |Gi|ψi ∈ HG, whereas the undefined parts ofUˆ transform|Gψiinto some

orthogonal state|G⊥_ψi ∈ HG⊥ of lesser interest.

In the following, we consider the case where Hˆ is a Hermitian matrix. The case of normal Hˆ is discussed inAppendix A. Thus the action ofUˆ on|GψiinEq. (2) can be written more clearly in the

eigenbasis ofHˆ|λi=λ|λi. For each eigenstate ofHˆ, ˆ

U|Gi|λi= ˆU|Gλi=λ|Gλi+

p

1− |λ|2_|_G⊥

λi, (3)

and we find it convenient to define the subspaceHλ=span{|Gλi,Uˆ|Gλi}. Note the trivial caseλ= 1

whereHλ is one-dimensional, which we will ignore. We also emphasize that the action ofUˆ on states

orthogonal to|Gλiare generally not easily controlled by the user, and thus left undefined.

Given any scalar function f, our goal is to find an optimal quantum circuit that transforms any standard-form encoding ofHˆ into a standard-form encoding off[ ˆH]≡P

λf(λ)|λihλ|. We accomplish

this with the quantum signal processor.

Definition 2(Quantum Signal Processor). A quantum signal processor solves the following.

Inputs: • A Hermitian matrixHˆ with bounded spectral norm k_Hˆ_{k ≤}₁_.

• A function f : [−1,1] → D, where D is the complex unit disc, i.e. ∀x ∈

[−1,1], |f(x)| ≤1.

Resources: • An encoding ofHˆ =hG|Uˆ|Giby the oracles of Definition 1.

• A constant number of additional qubits.

• Arbitrary single and two-qubit gates.

Outputs: • A standard-form encoding off[ ˆH], i.e. a unitaryQˆ wheref[ ˆH] =h0|Qˆ|0i.

In the simplest case, one might wish to applyHˆ multiple times to generate higher moments. For instance, Hˆ2 _{would allow a direct estimate of variance. Unfortunately, the subspace} _H

λ for each

eigenstate|λiis not invariant underUˆ in general. As a result, repeated applications in this basis do not produce higher moments ofHˆ due to leakage out of Hλ. The structure of this leakage depends

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Order can be restored to this undefined behavior by stemming the leakage. The simplest possibility that preserves the signal operator ofEq. (2)replacesUˆ with a unitary, theiterateWˆ, that also encodes

ˆ

H =hG|Wˆ|Gi, but for each eigenstate |λi, performs a rotation in SU(2)on disjoint two-dimensional subspacesHλ=span{|Gλi,Wˆ|Gλi}=span{|Gλi,|Gλ⊥i}. This defines the state|G⊥λithrough

Gram-Schmidt orthogonalization, and a set of Pauli operatorsXˆλ,Yˆλ,Zˆλ for each subspace.

|G⊥_λi= ( ˆW_p−λ)|Gλi 1− |λ|2 ,

ˆ

Xλ|Gλi=|Gλ⊥i, Yˆλ|Gλi=i|G⊥λi, Zˆλ|Gλi=|Gλi. (4)

In this basis, the iterate for each eigenvalue ofHˆ is defined to be exactly

ˆ

W =

λ −p

1− |λ|2

p

1− |λ|2 _λ

λ

= λ|GλihGλ| − p

1− |λ|2_|_G λihG⊥λ|

+p1− |λ|2_|_G⊥

λihGλ| +λ|G⊥λihG ⊥ λ|

, (5)

On a general input|GiP

λaλ|λi ∈ HG, the iterate is represented by

ˆ

W =M

λ

λ −p

1− |λ|2_.

p

1− |λ|2_. _λ

λ

=M

λ

e−iYˆλθλ_, ₍₆₎

whereθλ is defined through the equality

∀λ∈[−1,1], λ−ip1− |λ|2_{= cos (}_θ

λ)−isin (θλ)⇒θλ= cos−1(λ). (7)

In the following, the iterate will always be applied to states spanned by the subspaceL

λHλ. Thus

its action on states outside this subspace need not be defined. The usefulness of this construct is evident. Due to its invariant subspace, multiple applications of the iterate result in highly structured behavior. However, implementingWˆ appears difficult in general. One one hand, the functionp1− |λ|2

must be computed for each eigenstate. In principle, this can be approximated using phase estimation with considerable overhead [32]. On the other hand, it is not at all clear whether a unitary with the form of Wˆ can always be engineered from the standard-form encoding of Hˆ. ‘Qubitization’ is our solution to constructing this iterate with minimal overhead.

Theorem 2(Qubitization). Given a Hermitian matrix Hˆ =hG|aUˆ|Gia encoded in standard-form as

described inDefinition 1, the iterateWˆ ofEq.(6)can be constructed using at most one query each to

ˆ

G, controlled-U, their inverses, at most one additional qubit, and O(log (dim (Ha))) quantum gates.

Using the oraclesG,ˆ Uˆ, and arbitrary unitary operations ononlythe ancilla register, we also provide necessary and sufficient conditions inLemma 8, for whenWˆ can be implemented exactly using only one query toUˆ. As these conditions are somewhat restrictive, we then prove inLemma 10that qubitization isunconditionallypossible by instead using thecontrolled-Uˆ oracle in a quantum circuit that generates the sameHˆ and satisfies these conditions. We describe a similar construction for normal operators in Appendix.A.

Observe thatWˆN _{efficiently produces Chebyshev polynomials}_T

N[ ˆH] [9]. We call any function [·]

of the signalHˆ target operators when they occur in the top-left block and are thus automatically in standard-form. The fact that Chebyshev polynomials are the best polynomial basis forL∞ function

approximation on an finite interval [33] suggests that the any target operator f[ ˆH] =A[ ˆH] +iB[ ˆH] could be approximated with a judicious choice of controls on the ancilla register. We present two implementation of the quantum signal processor described inDefinition 2. The first does not require any additional qubits beyond that for qubitization, and realizes a broad class of target operatorsf.

Theorem 3 (Ancilla-Free Quantum Signal Processing). A quantum signal processor can be

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Inputs • The iterateWˆ obtained from the qubitization procedure ofTheorem 2on a Hermitian matrixHˆ =hG|Uˆ|Gi encoded by the oracles ofDefinition 1.

• Real polynomialsA(λ), B(λ)of degreeQand equal parity satisfying all following.

•A(1) = 1;

• ∀λ∈[−1,1], A2₍_λ_{) +}_B2₍_λ₎_≤₁_; • ∀λ≥1, A2₍_λ_{) +}_B2₍_λ₎_≥₁_;

• ∀Qeven, λ≥0, A2₍_iλ_{) +}_B2₍_iλ₎_≥₁_.

Output: • A standard-form encoding ofA[ ˆH] +iB[ ˆH]. Cost • Qqueries toWˆ.

• Zero additional qubits beyond that forWˆ.

• O(Qlog (dim (Ha)))arbitrary single and two-qubit gates.

Note that the two polynomials in our solution of ancilla-free quantum signal processing are of the same parity. At this point, we assume that allUˆ have been qubitized and so use the iterateWˆ as our basic building block. A different basis set of functions with fewer restrictions on parity can be obtained by embeddingWˆ into yet another SU(2)invariant subspace by adding an ancilla qubit. This leads to the second implementation of the quantum signal processor ofDefinition 2.

Theorem 4 (Single-Ancilla Quantum Signal Processing). A quantum signal processor can be

imple-mented with the following properties.

Inputs: • The iterateWˆ obtained from the qubitization procedure ofTheorem 2on a Hermitian matrixHˆ =hG|Uˆ|Gi encoded by the oracles ofDefinition 1.

• Real polynomialsA(λ), C(λ)of degreeQ/2and opposite parity satisfying all following.

•A(0) = 1;

• ∀λ∈[−1,1], A2₍_λ_{) +}_C2₍_λ₎_≤₁_;

Output: • A standard-form encoding ofA[ ˆH] +iC[ ˆH]. Cost: • Qqueries to controlled-Wˆ.

• One additional qubit beyond that forWˆ.

• O(Qlog (dim (Ha)))arbitrary single and two-qubit gates.

These powerful tools for target operator processing, made possible by qubitization, are agnostic to the underlying oracles that describe the signal operator Hˆ. In many instances, converting this description to the standard-form ofDefinition 1is straightforward and is indeed how our Hamiltonian simulation results for the varied input models ofTable 1are proven.

3 Explicit Encodings of

H

ˆ

= (

h

G

| ⊗

I

ˆ

) ˆ

U

(

|

G

i ⊗

I

ˆ

)

We now justify our motivation for encoding matricesHˆ in the standard-form format ofHˆ =hG|_Uˆ_|_G_i inDefinition 1. A number of common techniques for encoding matrix problems on quantum computers map naturally to the standard-form with minimal overhead. Thus taking the standard-form as our starting point is without any loss of generality. This is demonstrated by the following three explicit implementation of the oraclesUˆ andGˆ|0i=|Gifor Hamiltonians represented as a linear combination of unitaries,d-sparse Hamiltonians, and a new input model where Hamiltonians are represented by a purified density matrix.

3.1 Linear Combination of Unitaries

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on the fact that any complexHˆ is a linear combination of somedunitary operators:

ˆ

H =

d

X

j=1

αjUˆj, kHˆk ≤ k~αk1= d

X

j=1

|αj|, (8)

where the upper bound on the spectral norm iskα~k1. Note that this bound depends on the choice of

decomposition, but is tight for the some choice. Without loss of generality, allαj ≥0 by absorbing

complex phases intoUˆj. The algorithm assumes that the αj are provided as a list ofdnumbers, and

eachUˆj is provided as a quantum circuit composed of constant numberO(C)of primitive gates. With

these inputs, the oracles

ˆ G=

d

X

j=1

r _α

j k~αk1

|jih0|a+· · ·, Uˆ = d

X

j=1

|jihj|a⊗Uˆj, hG|Uˆ|Gi=

ˆ H

k~αk1

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can be constructed, where the ancilla state creation operator Gˆ|0i =|Gi is implemented with O(d) primitive gates, and the selectorUˆ is implemented withO(dC)primitive gate. By direct expansion of

ˆ

UGˆ|0ia|ψis, this leads exactly to Eq. (2). This proves the following.

Lemma 5(Standard-Form Encoding by a Linear Combination of Unitaries). LetGˆ prepare the state

|Gia = P d j=1

p

αj/k~αk1|jia where αj ≥ 0. Let Uˆ = P d

j=i|jihj|a ⊗Uˆj. These oracles encode the

matrixhG|Uˆ|Gi= _k_~_α1_k

1

Pd

j=1αjUˆj.

Proof. Consider the computation

(hG|a⊗Iˆs) ˆU(|Gia⊗Iˆs) = (hG|a⊗Iˆs)





d

X

j=1

r _α

j kα~k1

|jia⊗Uˆj



=

d

X

j,k=1 √

αjαk k~αk1

hk|ji ⊗_Uˆ_j

= 1

kα~k1 d

X

j=1

αjUˆj. (10)

Of course, the optimal decomposition that costs the fewest number of ancilla qubits and primitive gates may be difficult to find, and may not even fit naturally in this model, but LCU shows that implementing an encoding for anyHˆ is possible in principle.

3.2 d-Sparse Hamiltonians

The model ofd-sparse Hamiltonians is another paradigm for specifying matrices to quantum comput-ers, and is particularly common in the design of quantum algorithms by quantum walks [2]. Such Hamiltonians have at most d non-zero entries in any row, and information on their positions and matrix valuesHˆjk are accessed through the following two standard unitary oracles [4].

ˆ

OH|ji|ki|zi=|ji|ki|z⊕Hˆjki, OˆF|ji|li=|ji|f(j, l)i. (11)

Observe that OˆH accepts a rowj and columnk index and returns Hˆjk in some binary format. The

other oracle OˆF accepts a row j index and a number l ∈[d] to compute in-place the column index f(j, l)of thelth non-zero element in rowj. Given this input, we now encodeHˆ in standard-form.

Lemma 6(Standard-Form Encoding of ad-Sparse Hamiltonian). Let the oracles ofEq.(11)specify a

d-sparse Hamiltonian Hˆ with max-norm kHkmax. Then the oracles encodinghG|Uˆ|Gi= Hˆ

dk_Hˆ_k_max can

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Proof. Let |Gi=|0ia1|0ia2|0ia3 ≡ |0ia be a computational basis state, and let Fj ={f(j, l)}l∈[d] be

the set of column indices to all non-zero elements in rowj. Ref. [4] shows how each of the isometries

ˆ T1=

X

j

|ψjih0|ahj|s, |ψji=

X

p∈Fj |p_√ia3

d

s Hpj

kHˆkmax |0ia1+

s

1− |Hpj| kHˆkmax

|1ia1

|0ia2|jis, (12)

ˆ T2=

X

k

|χkih0|ahk|s, hχk|=

X

p∈Fk hp|s √

d

s Hkp

kHˆkmax h0|a2+

s

1− |Hkp| kHˆkmax

h1|a2

h0|a1hk|a3,

can be implemented using using 2 queries to ˆOH and 1 query to ˆOF. By construction, the overlap of

these states ishχk|ψji= Hkj dkHˆkmax

. Now choose ˆU = ˆT₂†Tˆ1. By a direct computation,

(hG|a⊗Iˆs) ˆU(|Gia⊗Iˆs) =

X

k,j

|kishχk|ψjihj|s=

X

k,j Hkj

dkHˆkmax

|kihj|s=

ˆ H

dkHˆkmax

. (13)

3.3 Purified Density Matrix

The simplicity of the standard-form encoding allows us to swiftly devise new input models for Hamil-tonians. Here, we consider the case where a Hamiltonian Hˆ = Tr [|GihG|]_a1 is a density matrix ρˆ obtained by tracing out the ancilla register of a pure state prepared by some oracleGˆ. That is,

ˆ

G|0ia =|Gia =

X

j √

αj|jia1|χjia2, ρˆ= Tr [|GihG|]a1=

X

j

αj|χjihχj|. (14)

Lemma 7 (Standard-Form Encoding of a Purified Density Matrix). Let the oracle of Eq.(14)

pre-pare a state Gˆ|0ia = |Gia = Pj √

αj|jia1|χjia2 that is a purification of the density matrix ρˆ =

P

jαj|χjihχj|= Tr [|GihG|]a1. Let

ˆ

U be a unitary that swaps the register a2 with the system register s. These oracles encode the matrixhG|Uˆ|Gi= ˆρ.

Proof. Let{|λi} be a complete basis on the system. By a direct computation,

hG|Uˆ|Gia

X

λ

|λihλ|s=

X

λ

X

j

hG|√αj|jia1|λia2|χjihλ|s=

X

λ

X

j

|αj||χjishχj|λihλ|s= ˆρ. (15)

4 Qubitization in a Quantum Signal Processor

This section describes qubitization in detail: the process for creating the iterateWˆ givenHˆ encoded in standard-form, and an essential component in a systematic procedure for implementing operator transformations ofHˆ. In Lemma 8, we provide necessary and sufficient conditions on when Wˆ can be constructed from the oracles G,ˆ Uˆ. Then in Lemma 10, we show that any G,ˆ Uˆ not satisfying these conditions can be efficiently transformed into a Gˆ0_,_Uˆ0 _{that do, and encode the same signal}

operatorHˆ =hG0|Uˆ0|G0i. Together, these Lemmas in the remainder of this section complete the proof ofTheorem 2for the case of HermitianHˆ.

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necessary and sufficient conditions on whatSˆ0 must be. As Sˆ0 is otherwise arbitrary, we use without loss of generality the ansatz of Sˆ0 being a product of a reflection about |Gi and another arbitrary unitarySˆon the ancilla:

ˆ

W = ((2|GihG| −Iˆ)a⊗Iˆs) ˆSU ,ˆ |Gλi=|Gi|λi ⇒ |G⊥λi=

λ|Gλi −SˆUˆ|Gλi √

1−λ2 . (16)

Lemma 8(Conditions on Qubitization). For all signal oraclesUˆ that implement the Hermitian signal

operatorHˆ, the unitarySˆ inEq.(16)creates a unitary iterateWˆ with the same signal operator in the same basis, but in an SU(2)invariant subspace containing |Giif and only if

hG|aSÛˆ|Gia = ˆH andhG|aSÛˆSÛˆ|Gia= Î. (17)

Proof. In the forward direction, we assume Eq. (6), then compute and compare with Eq. (16): λ=

hGλ|Wˆ|Gλi=hGλ|SˆUˆ|Gλi. By using this result repeatedly together with the fact that ˆSUˆ is unitary,

Gram-Schmidt orthonormalization of ˆW|Gλifurnishes the state|G⊥λi=

λ|Gλi−_Sˆ_Uˆ_|_Gλ_i √

1−λ2 which is

orthog-onal to|Gλi. By similarly computing and comparing− √

1−λ2₌_h_G

λ|Wˆ|G⊥λi= λ2_−h_G

λ|( ˆSU)ˆ 2|Gλi

√

1−λ2 , we

obtainhGλ|( ˆSUˆ)2|Gλi= 1. As these must be true for all eigenvectors |λi, the conditions inEq. (17)

are necessary.

That these are also sufficient follows from assumingEq. (17)and attempting to recover the compo-nents ofEq. (6)using the definitions ofEq. (16). By applyinghG|aSˆUˆ|Gia = ˆH, we compute ˆW|Gλi=

1−λ2 (2λ|Gλi −SˆUˆ|Gλi) =

2λ2−_√2λ2−λ2+1 1−λ2 =

√

1−λ2_.

A similar calculation for the remaining components requires hG|aSÛˆSÛˆ|Gia = Î and reveals that hG⊥_λ|_Wˆ_|_G⊥

λi = λ and hGλ|Wˆ|G⊥λi = − √

1−λ2_. _{As this must be true for all} _λ_{, we may indeed}

represent ˆW =L

λ

λ −√1−λ2

√

1−λ2 _λ

λ

.

In hindsight, these results are manifest. After all,hG|SÛˆSÛˆ|Gi= Î implies thatSÛˆ is a reflection when controlled by input state|Gi, and it is well-known that a Grover iterate [34,35] is the product of two reflection about start and target subspaces. Nevertheless, the sufficiency of these conditions highlights that this is the simplest method to extract controllable and predictable behavior out ofUˆ. In particular, these conditions are automatically satisfied in the trivial case withSˆ= ÎaswhenUˆ only

has eigenvalues±1, such as when it is a controlled-Pauli operator.

Corollary 9 (Qubitization of Reflections). For all signal oracles Uˆ that satisfyUˆ2 _{= ˆ}_I

as and

imple-ment the Hermitian signal operatorhG|aUˆ|Gia = ˆH, the unitary iterateWˆ = ((2|GihG| −Iˆ)a⊗Iˆs) ˆU

implements the same signal operator in the same basis, but in an SU(2) invariant subspace containing

|Gi.

Proof. Set ˆS= ˆIas inLemma 8, and verify thatEq. (17)is satisfied.

Unfortunately, a solution to Eq. (17) may not exist for more general Uˆ. Lemma 8, amounts to choosing Sˆ such that SˆUˆSˆ is the inverse Uˆ† whist preserving the signal operator hG|SˆUˆ|Gi = ˆH. Given thatSˆ only acts on the ancilla register, it is hard to see how this is always possible. Even if so,

ˆ

Smay be difficult to implement as it is an arbitrary unitary acting on a potentially large ancilla register. The solution is to construct a different quantum circuit Uˆ0 that containsUˆ but still implements the same signal operator, and crucially always has a extremely simple solutionSˆ. We now show how this can be done in all cases using only1query to controlled-Uˆ and controlled-Uˆ†.

Lemma 10 (Existence of Qubitization). For all signal unitaries Uˆ that implement the Hermitian

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controlled-Uˆ and controlled-Uˆ† once to implements the same signal operator. Moreover, Uˆ0 satisfies the conditionsEq.(17)

Proof. We prove this by an explicit construction. Let the controlled- ˆU operators be ˆV1=|0ih0| ⊗Iˆ+ |1ih1| ⊗Uˆ†_{, ˆ}_V

2 =|0ih0| ⊗Uˆ +|1ih1| ⊗Iˆ. Thus the extra qubit states |0i,|1i, are flags that selects

either Ûj or Ûj†. By multiplying, Û0= ˆV1Vˆ2=|0ih0| ⊗Uˆ +|1ih1| ⊗Uˆ†. Now consider the ancilla state |G0i=√1

2(|0i+|1i)|Gi, and choose ˆS= (|0ih1|+|1ih0|)⊗

ˆ

Ias. It is easy to verify that the conditions

ofEq. (17)is satisfied.

hG0|SˆUˆ0|G0i=hG0|Uˆ0|G0i=1

2( ˆH+ ˆH

†_{) = ˆ}_H, _h_G0_|_S_ˆ_U_ˆ0_S_ˆ_U_ˆ0_|_G0_i₌_h_G0_|_U_ˆ0†_U_ˆ0_|_G0_i_{= ˆ}_I_, ₍₁₈₎

where we have used the fact that ˆS|G0i=|G0i is an eigenstate, and that ˆS swaps the |0i,|1iancilla states in ˆU0, thus transforming it into its inverse.

Even if we are givenUˆ for which there is no solution toEq. (17), we can always applyLemma 10to construct aUˆ0that does with minimal overhead. Furthermore our proof uses no information about the detailed structure of|Gi. Thus without loss of generality, we can assume that any G,ˆ Uˆ have already been qubitized.

5 Operator Function Design on a Quantum Signal Processor

The purpose of the quantum signal processor is to transform the signalHˆ into any desired target op-eratorf[ ˆH]. We present a systematic framework that furnishes the optimal complexity and a concrete procedure for almost anyf and show how an exact connection is made between query complexity and the theory of best function approximations with polynomials [33,36].

Qubitization inSection 4is the essential first step that makes this endeavor plausible, as evidenced by the highly structured behavior of the iterate Wˆ in Eq. (6), where for Hermitian Hˆ, multiple applications elegantly generate Chebyshev polynomials TL[ ˆH] [9]. To go further, additional control

parameters onWˆ are necessary, and in the following, we only consider HermitianHˆ. Thus we introduce thephased iterate Wˆφ with the same invariant subspace asWˆ, and parameterized byφ∈R.

ˆ Wφ=

M

λ

λ −ie−iφp

1− |λ|2 −ieiφp1− |λ|2 _λ

λ

=M

λ

e−iφˆλθλ_, ₍₁₉₎

whereφˆλ_{= cos (}_φ_{) ˆ}_X

λ+ sin (φ) ˆYλ, and the eigenphaseθλis defined similar toEq. (7).

Lemma 11. The phased iterate Wˆφ in Eq. (19) is equal to Wˆφ = ˆZφ+π/2WˆZˆ−φ−π/2, where Zˆφ =

ˆ Zφ=

M

λ

e−iφ ₀

0 −1

λ

. (20)

λiand ˆZφ|G⊥λi=−|G⊥λi. As this true for allλ,Eq. (20)follows. Combining the representation

of ˆW fromEq. (6)with this leads toEq. (19),

ˆ

Wφ= ˆZφ−π/2WˆZˆ−φ+π/2=

M

λ

−ie−iφ ₀

0 −1

λ

λ −p

1− |λ|2

p

1− |λ|2 _λ

λ

ieiφ ₀

0 −1

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We provide algorithms with different overheads for large classes of transformations. ‘Ancilla-free quantum signal processing’ inSection 5.1 implement target operators where the real and imaginary parts have the same parity with respect toHˆ. Opposite parity is obtained by ‘single-ancilla quantum signal processing’ inSection 5.2.

5.1 Ancilla-Free Quantum Signal Processing

Consider a sequence ofQof phased iterates, where the angleφdefining each may differ.

ˆ

W_φ~ = ˆWφQ· · ·Wˆφ2Wˆφ1 =

M

λ

e−iφˆλQθλ· · ·_e−iφ λ

2θλ_e−iφˆ

λ

1θλ_, _φˆλ

j = cos (φj) ˆXλ+ sin (φj) ˆYλ.

=M

λ

A(2θλ)ˆIλ+iB(2θλ) ˆZλ+iC(2θλ) ˆXλ+iD(2θλ) ˆYλ. (21)

In each subspaceHλ, this is a product of SU(2) rotations. As such, we may decompose this in the

Pauli basisIˆλ,Xˆλ,Yˆλ,Zˆλ with the real functions (A,B,C,D)as coefficients. As we can only prepare

between elements of the sequence, leading to(A,B,C,D)with some non-trivial functional dependence onHˆ. Though the dependence of the output onφ~ seems hard to intuit, they nevertheless specify a

program for computing functions ofHˆ, similar to how a list of numbers might specify a polynomial. To understandEq. (21), it suffices to study the following sequence of single-qubit rotations.

ˆ Uφ~=e

−iφˆQθ/2_{· · ·}_e−iφˆ2θ/2_e−iφˆ1θ/2₌_A₍_θ_)ˆ_I₊_i_B₍_θ_{) ˆ}_Z₊_i_C₍_θ_{) ˆ}_X₊_i_D₍_θ_{) ˆ}_{Y ,} ₍₂₂₎

These functions previously characterized in Ref. [37], and found the following.

Lemma 12(AchievableA, B – Thm. 2.3 of Ref. [37]). For any integer Q >0, a choice of functions

A(θ)≡A(x),B(θ)≡B(x)inEq.(22)is achievable by someφ~∈RQ _{if and only if all the following are}

true:

(1)A(x)and B(x)are real parity-(Q mod 2)polynomials inx= cos (θ/2) of degree at mostQ; (2)A(1) = 1;

(3)∀x∈[−1,1],A2₍_x_{) +}_B2₍_x₎_≤₁_;

(4)∀x≥1,A2₍_x_{) +}_B2₍_x₎_≥₁_;

(5)∀Qeven, x≥0,A2₍_ix_{) +}_B2₍_ix₎_≥₁_.

Moreover,φ~∈RQ _{can be computed in classical}_O₍_poly₍_Q₎₎_time.

In other words, we may specify only the componentsA,B, independent of the othersC,Dthat are of lesser interest. By mapping this result toEq. (21), we complete the proof ofTheorem 3.

Proof ofTheorem 3. The unitary operators in each subspaceHλ of ˆW~φare isomorphic to the product

of single qubit rotationse−iφˆQθ/2· · ·_e−iφˆ2θ/2_e−iφˆ1θ/2_{. We identify} _θ/_{2 =}_θ

λ, where θλ = cos−1(λ) as

defined inEq. (7). Thus the result follows fromLemma 12by substitutingx= cos (θλ) =λ.

As polynomials form a complete basis on bounded real intervals, these results imply the query complexity of approximating any real functionA[ ˆH]with erroris exactly that of its best polynomial -approximation satisfying the constraints ofTheorem 3, and similarly for the complex case.

5.2 Single-Ancilla Quantum Signal Processing

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Ref. [7]. Given any unitaryVˆ with eigenstatesVˆ|λi=eiθλ_|_λ_i_and _Vˆ

0=|+ih+|b⊗Iˆs+|−ih−|b⊗Vˆ

controlled by the single-qubit ancilla registerbwhere Xˆb|±ib=±|±ib, consider the sequence

ˆ Vϕ~ =

Q/2

Y

kodd≥1

ˆ Vϕ†k+1+π

ˆ Vϕk = ˆV

† ϕQ+π· · ·

ˆ Vϕ3Vˆ

† ϕ2+π

ˆ

Vϕ1, Vˆϕ= (e

−iϕZ/2ˆ

⊗Iˆs) ˆV0(eiϕ ˆ Z/2

⊗Iˆs). (23)

For each eigenstate|λi, we obtain a product of single qubit operators e−iϕˆQθλ/2_{· · ·}_e−iϕˆ1θλ/2 _similar

toEq. (21) but with a halved phased, and these only act on the ancillab. Using the same reasoning as inSection 5.1, the choice ofϕ~ determines the effective single-qubit ancilla operator

ˆ Vϕ~ =

M

λ

ˆ

IbA(θλ) +iZˆbB(θλ) +iXˆbC(θλ) +iYˆbD(θλ)

⊗ |λihλ|s. (24)

In the following, we focus on the functionsA,C, which we also characterized fully in previous work.

Lemma 13(Achievable (A,C) – Thm. 1 of Ref. [7]). For any even integerQ >0, a choice of functions

A(θ),C(θ)in Eq.(22)is achievable by someφ~∈RQ _{if and only if all the following are true:}

(1)A(θ) =PQ/2

k=0akcos (kθ)is a real cosine Fourier series of degree at most Q/2;

(2)C(θ) =PQ/2

k=1cksin (kθ)is a real sine Fourier series of degree at mostQ/2;

(3)A(0) = 1;

(4)∀θ∈R,A2₍_θ_{) +}_C2₍_θ₎_≤₁_.

Moreover,φ~∈RQ _{can be computed in classical}_O₍_poly₍_Q₎₎_time.

In some cases, one might be given target functionsA,C that are only-close to being achievable, for instance, ifA,Care the output of some numerical procedure. This poses no fundamental difficulty, as we prove in the following, which generalizes slightly a similar statement in Ref. [7].

Lemma 14 (Stability of Achievable (A,C)). For any even integerQ >0, let

(1)A˜(θ) =PQ/2

k=0akcos (kθ)be a real cosine Fourier series of degree at mostQ/2;

(2)C˜(θ) =PQ/2

k=1cksin (kθ)be a real sine Fourier series of degree at mostQ/2;

(3)A˜(0) = 1 +1, where1≤1;

(4)∀θ∈R,A˜2₍_θ_{) + ˜}_C2₍_θ₎_≤_{1 +}

2, where2∈[0,1].

Then there exists Fourier seriesA,C that satisfy the conditions ofLemma 13, approximate

max

θ∈R |(A+iC)−( ˜A+i

˜

C)|=O(p|1|+2), (25)

and are computable in classicalO(poly(Q))time.

Proof. First, we satisfy condition (4) of Lemma 13 by rescalingA1= ˜ A

1+2, C=

˜ C 1+2.

Second, we use the polynomial sum-of-squares technique to compute real Fourier series B1 =

PQ/2

k=0bkcos (kθ) andD1=

PQ/2

k=1dksin (kθ) such thatA 2

1+B12+C2+D12= 1. Following [37, Section

C], this can be done in classical O(poly(Q)) time. By construction, A2

1(0) +B21(0) = 1 asC andD1

are odd functions. Let us define sin (δ)≡ B1(0). Hence, cos (δ)≡ A1(0) = 1+₁₊1

2.

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Last, we bound the error of this approximation.

max

θ∈R |(A+iC)−( ˜A+i

˜

C)|

≤max

θ∈R |(A+iC)−(A1+iC)|+ maxθ∈R |(A1+iC)−( ˜A+i

˜

C)| by a triangle inequality

≤max

θ∈R |(cos (δ)−1)A1(θ) + sin (δ)B1(θ)|+2

≤

q

(cos (δ)−1)2_{+ sin}2₍_δ_{) +}

2 usingA21(θ) +B21(θ)≤1

≤√2p1−cos (δ) +2=

r

22−1 1 +2

+2

=O(p|1|+2). (26)

We now proveTheorem 4 by evaluatingEq. (24)usingLemma 13for the case where the arbitrary unitaryVˆ is replaced with the more structured iterateeiΦ_Wˆ_{, and also where we have added a global} phaseΦ∈R.

Proof ofTheorem 4. Observe from Eq. (6) that eiΦ_Wˆ _{can be diagonalized to obtain its eigenvalues} θλ± and eigenvectors|Gλ±i.

eiΦWˆ|Gλ±i=ei(Φ+θλ±)|Gλ±i, θλ±=∓cos−1(λ), |Gλ±i=

|Gλi ±_√i|G⊥λi

2 . (27)

With this substitution ˆV ←eiΦ_W_ˆ_, _{Eq. (24)}_becomes

ˆ Vϕ~ =

M

λ,±

ˆ

IbA(Φ +θλ±) +iXˆbC(Φ +θλ±) +· · ·

⊗ |Gλ±ihGλ±|as. (28)

in

hG|ah+|bVˆϕ~|+ib|Gia =

M

λ,±

(A(Φ +θλ±) +iC(Φ +θλ±))⊗ hG|a|Gλ±ihGλ±|as|Gia

=M

λ

P

±(A(Φ +θλ±) +iC(Φ +θλ±))

2 ⊗ |λihλ|. (29)

For this proof, it suffices to choose Φ =π/2, and setak = 0 for all oddk, andck = 0 for all even k. We then evaluate each component of Eq. (29) using Lemma 13 and the Chebyshev polynomials Tk(x)≡cos (kcos−1(x)).

A(π/2 +θλ±) = Q/2

X

keven

akcos (k(π/2 +θλ±)) = Q/2

X

keven

akikTk(λ) = Q/2

X

keven

a0_kTk(λ) =A(λ), (30)

C(π/2 +θλ±) = Q/2

X

kodd

cksin (k(π/2 +θλ±)) = Q/2

X

kodd

ckik−1Tk(λ) = Q/2

X

kodd

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ThusEq. (29)simplifies to

hG|ah+|bVˆϕ~|+ib|Gia =

M

λ

(A(λ) +iC(λ))⊗ |λihλ|. (31)

We now express the conditions of Lemma 13in terms of polynomials. AsTk(−x) = (−1)kTk(x),

conditions (1) and (2) map to A(λ) being an even polynomial and C(λ) being an odd polynomial respectively. Whenθλ± =−π/2, this implies thatλ= cos (±π/2) = 0. Hence condition (3) maps to A(π/2 +θλ± = 0) = 1⇒ A(λ= 0) = 1. Asλ∈ [−1,1] for all values ofθλ±, condition (4) maps to ∀λ∈[−1,1], A2₍_λ_{) +}_C2₍_λ₎_≤_1.

Of course, other choices of Φ, such as Φ = 0, π/3, π/4, ..., lead to different families of target functions. However, those are beyond the scope of this work.

6 Application to Hamiltonian Simulation

With these results for qubitization and operator function design with a quantum signal processor, the application to Hamiltonian simulation follows easily. We complete the proofTheorem 1, and then apply these results to obtain our claims of improvements inTable 1. Given any HamiltonianHˆ encoded in the standard-form of Definition 1, Theorem 4 allows us to use 2Q queries to encode exactly any functionA[ ˆH] +iC[ ˆH]where A(λ)and C(λ)are bounded polynomials of opposite parity and degree Q. Hamiltonian simulation is accomplished by choosing a good degree Q polynomial approximation toA(λ) +iC(λ) =e−iλt_.

Proof ofTheorem 1. Similar to previous approaches [6, 7], we approximate e−iλt with the Jacobi-Anger expansioneicos (z)t=P∞

k=−∞i k_J

k(t)eikz [38], whereJk(t) are Bessel function of the first kind.

By identifying cos (z) =λ, suitable polynomialsA, Care obtained by a truncation and rescaling of

e−iλt=J0(t) + 2 ∞

X

keven>0

(−1)k/2Jk(t)Tk(λ) +i2 ∞

X

kodd>0

(−1)(k−1)/2Jk(t)Tk(λ). (32)

The error from truncating this expansion fork > Q=q−1 is a sum of|Jk(t)|that was bounded in [6]:

= max

λ∈[−1,1]

2

∞

X

k=q

(−1)k2J_k(t)T_k(λ)

≤ ∞

X

k=q

2|Jk(t)| ≤

4tq

2q_q_! =O

et q

q

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⇒log ₁

=O

qlog

₂_q

et

⇒q=O(t+ log (1/)).

The rapid converge by truncation arises ase−iλt_{is an entire analytic function [}₃₉_].

With the choice

˜

A(λ) =J0(t) + 2 Q

X

keven>0

(−1)k/2Jk(t)Tk(λ), C˜(λ) = 2 Q

X

kodd>0

(−1)(k−1)/2Jk(t)Tk(λ) (34)

the error maxλ∈[−1,1]|A˜(λ) +iC˜(λ)−e−iλt| ≤. Though this choice satisfies conditions (1-2) of Theo-rem 4, it is only-close to satisfying conditions (3-4). Fortunately, this is not a fundamental problem following the stability analysis ofLemma 14. One can perturb ˜A(λ),C˜(λ) to construct approximations A(λ) +iC(λ) that are achievable, with error maxλ∈[−1,1]|A(λ) +iC(λ)−e−iλt| =O(

p

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kh+|bhG|aVˆϕ~|Gia|+ib−e−i ˆ

Ht_k ₌_O₍√_{). The failure probability is}_O₍_{), and solving for}_Q _furnishes

stated the number of queries to ˆW and hence the oracles encoding ˆH=hG|Uˆ|Gi.

As we have already shown in Section 3 how a variety of matrix input models such as linear-combination-of-unitaries,d-sparse Hamiltonians, and purified density matrices map to the standard-form encoding, their respective algorithms for simulating the time-evolution operators follow trivially.

Corollary 15 (Hamiltonian Simulation of a Sparse Hermitian Matrix). Given access to the oracle

ofEq.(11)specifying a d-sparse Hamiltonian Hˆ, time evolution byHˆ can be simulated for timet and errorwithO(dtkHˆkmax+ log (1/))queries toOˆH,OˆF.

Proof. Follows from combining Theorem 1with the standard-form encodingLemma 6.

The query complexity in Eq. (33) for this sparse case exactly matches a lower bound based on simulating a Hamiltonian that solves PARITY with unbounded error [21], valid for all parameter values, and not just in asymptotic limits [6,7]. This completes the optimality proof ofTheorem 1.

The case whereHˆ decomposes into a linear combination of unitaries is an immediate application:

Corollary 16 (Hamiltonian Simulation of a Linear Combination of Unitaries). Given access to the

oracles ofEq.(9)specifying a HamiltonianHˆ =Pd

j=1αjUˆj, time evolution byHˆ can be simulated for

timetand error withO(αt+ log (1/))queries.

The intuitiveness ofTheorem 1allows us to swiftly devise new models of Hamiltonian simulation.

Corollary 17 (Hamiltonian Simulation of a Purified Density Matrix). Given access to the

ora-cle Eq. (14) specifying a Hamiltonian Hˆ = ˆρ that is a density matrix ρˆ, time evolution by Hˆ can be simulated for timet and errorwith O(t+ log (1/))queries.

In all the above, the query complexityO(· · ·t+log (1/))is an upper bound on the more precise form ofEq. (33). The exact the tradeoff between, t, Qin Eq. (33)is studied numerically in Appendix B, together with example phasesϕ~ implementing Vˆϕ~ for the polynomials inEq. (34).

7 Conclusion

Our general procedure for Hamiltonian simulation inTheorem 1extends the scope of possible useful formulations of Hamiltonian simulation. As seen in Table 1, it encompasses any case where the Hamiltonian is embedded in a flagged subspace of the signal unitary. Given this, a simulation algorithm with query complexity optimal in all parameters, and also not just in asymptotic limits, is easily obtained with minimal overhead. While this procedure contains and significantly improves upon important models where the Hamiltonian is d-sparse or a linear combination of unitaries, its greater value lies in illuminating an intuitive and straightforward path to other as-yet undiscovered models of Hamiltonian simulation. In particular, our result for time-evolution by a purified density matrix is a quadratic improvement in time and an exponential improvement in error over the sample-based model – the proof of which consisted of just a few lines.

Many other exciting directions extend from this work. One example is how additional structural information aboutHˆ [40] may be exploited. This is illustrated by when the spectral norm ofkHˆkis significantly smaller than the sum of coefficientsk~αk1 of a particular linear combination of unitaries

decomposition. If this decomposition were to be used, simulation would take timeO kα~k1t+ log (1/)

Hamiltonian Simulation by Qubitization