Translationally Invariant Universal Models in 1D
In this section, we prove our main result: there exist translationally invariant, nearest neighbor Hamiltonians acting on a chain of qudits, which are universal quantum simulators.
All the ‘circuit-to-Hamiltonian’ mappings we make use of in this work are what are known as ‘standard form Hamiltonians.’ Where ‘Standard form Hamiltonians’ are a certain class of circuit-to-Hamiltonian constructions, defined in [22]. We refer interested readers to [22] for the full definition—and simply note that it encompasses the Turing-machine-based mappings which we make use of in this work [14, 15]. In [22], the following result was shown, which we will make use of in our proofs:
Lemma 4.1
(Standard form ground states; restatement of [22, Lem. 5.8, Lem. 5.10]). Let \(H_\mathrm {SF}\) be a Standard Form Hamiltonian encoding a computation U, which takes (classical) inputs from a Hilbert space \({\mathcal {S}}\), and which sets an output flag with certainty if it is given an invalid input. For \(\mathinner {|{\psi _\mu }\rangle } \in {\mathcal {S}}\) and \(\Pi _{t=1}^T U_t = U\) we define
$$\begin{aligned} \mathinner {|{\Phi (U,\psi _\mu )}\rangle } {:}{=}\frac{1}{\sqrt{T}} \sum _{t=1}^{T} U_t\ldots U_1 \mathinner {|{\psi _\mu }\rangle }\mathinner {|{t}\rangle }. \end{aligned}$$
Then, \({\mathcal {L}} = \text {span }\{ \mathinner {|{\Phi (U,\psi _\mu )}\rangle } \}_{\mu =1}^{d^n}\) defines the kernel of \(H_{SF}\), i.e., \(H_\mathrm {SF}|_{{\mathcal {L}}} = 0\). The smallest nonzero eigenvalue of \(H_\mathrm {SF}\) scales as \(1 - \cos {\pi / 2T}\).
We also require a digital quantum simulation algorithm, summarized in the following lemma:
Lemma 4.2
(Implementing a Local Hamiltonian Unitary). For a k-local Hamiltonian \(H=\sum _{i=1}^m h_i\) on an n-partite Hilbert space of local dimension d, and where \(m={{\,\mathrm{poly}\,}}n\), there exists a QTM that implements a unitary \({{\tilde{U}}}\) such that
$$\begin{aligned} {{\tilde{U}}} = \mathrm {e}^{\mathrm {i}H t} + {{\,\mathrm{O}\,}}(\epsilon ), \end{aligned}$$
and which requires time \({{\,\mathrm{poly}\,}}(1/\epsilon , d^k, \Vert H \Vert t, n)\).
Proof
Follows directly from [23, 24]. \(\square \)
The polynomial time bound in Lemma 4.2 suffices for our purposes; a tighter (and more complicated) bound, also for the more general case of sparse Hamiltonians, can be found in [25].
We can now start our main analysis by proving that ‘dovetailing’ quantum computations—rigorously defined and constructed in [14, Lem. 22]—can be used to construct universal simulators.
Lemma 4.3
(Dovetailing for simulation). Let \(M_1\) be a QTM which writes out the binary expansion of some \(x \in \mathbb {N}\) on its work tape. Assume there exists a standard form Hamiltonian which encodes the Turing machine \(M_1\). Then, there also exists a standard form Hamiltonian \(H_{\mathrm {SF}}(x)\), which encodes the computation \(M_1\) dovetailed with a QTM \(M_\mathrm {PE}\), such that the family of Hamiltonians
$$\begin{aligned} H_{\mathrm {univ}}(x) = \Delta H_{\mathrm {SF}}(x) + T \sum _{i=0}^{N-1} \left( \sqrt{2} \Pi _{\alpha } - \Pi _{\beta } \right) \end{aligned}$$
(9)
can simulate any quantum Hamiltonian. Here \(\Delta \) and T are parameters of the model, and \(\Pi _{\alpha }\) and \(\Pi _{\beta }\) are one-body projectors,
Proof of Lemma 4.3
To prove this, we show that the \(H_{\mathrm {univ}}(x)\) can satisfy the definition to be an approximate simulation of an arbitrary ‘target Hamiltonian’ \(H_{\mathrm {target}}\), to any desired accuracy. We break up the proof into multiple parts. First, we construct a history state Hamiltonian \(H_{\mathrm {SF}}(x)\), which encodes two Turing machine computations: \(M_1\) which extracts a description of \(H_{\mathrm {target}}\) from a parameter of \(H_{\mathrm {SF}}\), and \(M_\mathrm {PE}\) which carries out phase estimation on the unitary generated by \(H_{\mathrm {target}}\). Then, we define the one-body projectors \(\Pi _\alpha \) and \(\Pi _\beta \) which break up the ground space degeneracy of \(H_{\mathrm {SF}}\), and inflict just the right amount of penalty to approximately reconstruct the spectrum of \(H_{\mathrm {target}}\) in its entirety.
Construction of H\(_\mathbf{SF} \). \(H_{\mathrm {SF}}\) is a standard form history state Hamiltonian with a ground space laid out in Lemma 4.1. The local states of the spins on which \(H_{\mathrm {SF}}\) acts are divided into multiple ‘tracks.’ There are a constant number of these, hence a constant local Hilbert space dimension. The exact number will depend on the standard form construction being used. Each track serves its own purpose, as outlined in Table 1. See [14, 15] for more detail.
Table 1 Local Hilbert space decomposition for \(H_{\mathrm {SF}}\)
The QTM \(M_\mathrm {PE}\) reads in the description of \(H_{\mathrm {target}}\)—provided as integer \(x\in \mathbb {N}\) output by the Turing machine \(M_1\) whose worktape it shares. \(M_\mathrm {PE}\) further reads in the unconstrained input state \(\mathinner {|{\psi }\rangle }\) (see Table 1 for details of the local Hilbert space decomposition). But instead of proceeding immediately, \(M_\mathrm {PE}\) idles for L time-steps (where L is specified in the input string x, as explained in Sect. 3.2), before proceeding to carry out the quantum phase estimation algorithm.
The quantum phase estimation algorithm is carried out with respect to the unitary \(U = \mathrm {e}^{\mathrm {i}H_{\mathrm {target}}\tau }\) for some \(\tau \) such that \(\Vert H_{\mathrm {target}}\tau \Vert < 2\pi \). It takes as input an eigenvector \(\mathinner {|{u}\rangle }\) of U, and calculates the eigenphase \(\phi _u\). The output of \(M_\mathrm {PE}\) is then the pair of integers \((a_u,b_u)\) (corresponding to the extracted phase \(\phi _u=\sqrt{2} a_u - b_u\) as explained in Remark 3.1), specified in binary on an output track. To calculate \(\lambda _u\)—the eigenvalue of \(H_{\mathrm {target}}\)—to accuracy \(\epsilon \) requires determining \(\phi _u\) to accuracy \({{\,\mathrm{O}\,}}(\epsilon /\Vert H_{\mathrm {target}}\Vert )\) which takes \({{\,\mathrm{O}\,}}(\Vert H_{\mathrm {target}}\Vert /\epsilon )\) uses of \(U=\mathrm {e}^{\mathrm {i}H_{\mathrm {target}}\tau }\). The unitary U must thus be implemented to accuracy \({{\,\mathrm{O}\,}}(\epsilon / \Vert H_{\mathrm {target}}\Vert )\), which is done using Lemma 4.2; the latter introduces an overhead \({{\,\mathrm{poly}\,}}(n,d^k,\Vert H_{\mathrm {target}}\Vert ,\tau ,1/\epsilon )\) in the system size n, local dimension d, locality k, and target accuracy \(\epsilon \). The error overhead of size \({{\,\mathrm{poly}\,}}1/\epsilon \) due to the digital simulation of the unitary is thus polynomial in the precision, as are the \(\propto 1/\epsilon \) repetitions required for the QPE algorithm. The whole procedure takes time
$$\begin{aligned} T_\mathrm {PE}{:}{=}{{\,\mathrm{poly}\,}}(d^k, \Vert H_{\mathrm {target}}\Vert /\epsilon ,n). \end{aligned}$$
(10)
In our construction, the input to \(M_\mathrm {PE}\) is not restricted to be an eigenvector of \(\mathinner {|{u}\rangle }\), but it can always be decomposed as \(\mathinner {|{\psi }\rangle } = \sum _u m_u \mathinner {|{u}\rangle }\). By linearity, for input \(\mathinner {|{\psi }\rangle } = \sum _u m_u \mathinner {|{u}\rangle }\) the output of \(M_\mathrm {PE}\) will be a superposition in which the output \((a_u,b_u)\) occurs with amplitude \(m_u\).
After \(M_\mathrm {PE}\) has finished its computation, its head returns to the end of the chain. A dovetailed counter then decrements \(a_u, a_u-1, \ldots , 0\) and \(b_u, b_u-1, \ldots , 0\).Footnote 4 For each timestep in the counter \(a_u, a_u-1, \ldots , 0\) the Turing machine head changes one spin to a special flag state \(\mathinner {|{\Omega _a}\rangle }\) which does not appear anywhere else in the computation, while for each timestep in the counter \(b_u, b_u-1, \ldots , 0\) the Turing machine head changes one spin to a different flag state \(\mathinner {|{\Omega _b}\rangle }\). (See, e.g., [26, Lem. 16]) for a construction of a Turing machine with these properties.)
By Lemma 4.1, the ground space \({\mathcal {L}}\) of \(H_{\mathrm {SF}}\) is spanned by computational history states as given in Definition 2.7 and is degenerate since any input state \(\mathinner {|{\psi }\rangle }\) yields a valid computation. Therefore,
$$\begin{aligned} \mathrm {ker}(H_{\mathrm {SF}}) = {\mathcal {L}} = \mathrm {span}_{\mathinner {|{\psi }\rangle }}\left( \frac{1}{\sqrt{T}} \sum _{t=1}^{T} \mathinner {|{\psi ^{(t)}}\rangle }\mathinner {|{t}\rangle }\right) \end{aligned}$$
(11)
where \(\mathinner {|{\psi ^{(t)}}\rangle }\) denotes the state of the system at time step t if the input state was \(\mathinner {|{\psi }\rangle }\).
A Local Encoding. In order to prove that \(H_{\mathrm {univ}}(N)\) can simulate all quantum Hamiltonians, we need to demonstrate that there exists a local encoding \({\mathcal {E}}(M)\) such that the conditions of Definition 2.3 are satisfied. To this end, let
$$\begin{aligned} \mathinner {|{\Phi _\mathrm {idling}(\psi )}\rangle } {:}{=}\frac{1}{\sqrt{L'}} \sum _{t=1}^{L'} \mathinner {|{\psi ^{(t)}}\rangle }\mathinner {|{t}\rangle } \end{aligned}$$
where \(L' = T_1 + L\), and where \(T_1\) is the number of time steps in the \(M_1\) computation. This is the history state up until the point that \(M_\mathrm {PE}\) begins its computation (i.e., the point at which the ‘idling to enhance coherence’ ends). So, throughout the computation encoded by this computation the spins which encode the information about the input state remain in their initial state, and we can write:
$$\begin{aligned} \mathinner {|{\Phi _\mathrm {idling}(\psi )}\rangle } = \mathinner {|{\psi }\rangle } \otimes \frac{1}{\sqrt{L'}} \sum _{t=1}^{L'} \mathinner {|{t}\rangle } \end{aligned}$$
The rest of the history state we capture is
$$\begin{aligned} \mathinner {|{\Phi _\mathrm {comp}(\psi )}\rangle } {:}{=}\frac{1}{\sqrt{T - L'}} \sum _{t=L'+1}^{T} \mathinner {|{\psi ^{(t)}}\rangle }\mathinner {|{t}\rangle }, \end{aligned}$$
such that the total history state is
$$\begin{aligned} \mathinner {|{\Phi (\psi )}\rangle } = \sqrt{\frac{L'}{T}} \mathinner {|{\Phi _\mathrm {idling}(\psi )}\rangle } + \sqrt{\frac{T - L'}{T}} \mathinner {|{\Phi _\mathrm {comp}(\psi )}\rangle }. \end{aligned}$$
We now define the encoding \({\mathcal {E}}(M) = V M V^\dagger \) via the isometry
$$\begin{aligned} V = \sum _i \mathinner {|{\Phi _\mathrm {idling}(i)}\rangle }\mathinner {\langle {i}|}. \end{aligned}$$
(12)
where \(\mathinner {|{i}\rangle }\) are the computational basis states (any complete basis will suffice). \({\mathcal {E}}\) is a local encoding, which can be verified by a direct calculation:
$$\begin{aligned} \begin{aligned} {\mathcal {E}}(A_j \otimes \mathbb {1})&= \sum _{ik}\mathinner {|{\Phi _\mathrm {idling}(i)}\rangle } \mathinner {\langle {i}|}(A_j \otimes \mathbb {1})\mathinner {|{k}\rangle }\mathinner {\langle {\Phi _\mathrm {idling}(k)}|} \\&= \sum _{ik} \mathinner {|{i}\rangle }\mathinner {\langle {i}|}(A_j \otimes \mathbb {1}) \mathinner {|{k}\rangle }\mathinner {\langle {k}|} \otimes \frac{1}{L} \sum _{tt'=1}^L \mathinner {|{t}\rangle }\mathinner {\langle {t'}|} \\&= (A_j \otimes \mathbb {1}) \sum _{i} \mathinner {|{i}\rangle }\mathinner {\langle {i}|} \otimes \frac{1}{L} \sum _{tt'=1}^L \mathinner {|{t}\rangle }\mathinner {\langle {t'}|} \\&=\left( A^\mathrm {phys}_j \otimes \mathbb {1}\right) \sum _i\mathinner {|{\Phi _\mathrm {idling}(i)}\rangle }\mathinner {\langle {\Phi _\mathrm {idling}(i)}|} \\&= \left( A^\mathrm {phys}_j \otimes \mathbb {1}\right) {\mathcal {E}}(\mathbb {1}), \end{aligned} \end{aligned}$$
(13)
where \(A^\mathrm {phys}_j\) is the operator A acting on the Hilbert space corresponding to the \(j^{\hbox {th}}\) qudit.
We now consider the encoding \({\mathcal {E}}'(M) = V'MV^{\prime \dagger }\), defined via
$$\begin{aligned} V' = \sum _i \mathinner {|{\Phi (i)}\rangle }\mathinner {\langle {i}|}. \end{aligned}$$
(14)
We have that
$$\begin{aligned} \begin{aligned} \Vert V' - V\Vert ^2&= \left\| \sum _i \left( \mathinner {|{\Phi (i)}\rangle }\mathinner {\langle {i}|} - \mathinner {|{\Phi _\mathrm {idling}(i)}\rangle }\mathinner {\langle {i}|}\right) \right\| ^2\\&= \left\| \sum _i \left( \sqrt{\frac{T -L'}{T}}\mathinner {|{\Phi _\mathrm {comp}(i)}\rangle }\mathinner {\langle {i}|} + \left( \sqrt{\frac{L'}{T}}-1\right) \mathinner {|{\Phi _\mathrm {idling}(i)}\rangle }\mathinner {\langle {i}|} \right) \right\| ^2 \\&\le 2\left( 1-\sqrt{\frac{L'}{T}}\right) \le 2\frac{T-L'}{T}=2\frac{T_\mathrm {PE}}{T}. \end{aligned}\nonumber \\ \end{aligned}$$
(15)
By Lemma 4.1, \(S_{{\mathcal {E}}'}\) is the ground space of \(H_{\mathrm {SF}}\).
Splitting the Ground Space Degeneracy of H\(_\mathbf{SF }\). What is left to show is that there exist one body-projectors \(\Pi _{\alpha }\) and \(\Pi _{\beta }\) which add just the right amount of energy to states in the kernel \({\mathcal {L}}(H_{\mathrm {SF}})\) to reproduce the target Hamiltonian’s spectrum. We first choose the one body terms in \(H_{\mathrm {univ}}\) to be projectors onto local subspaces which contain the two states which are outputs of the \(M_\mathrm {PE}\) computation—\(\mathinner {|{\Omega _a}\rangle }\) and \(\mathinner {|{\Omega _b}\rangle }\):
We have shown that if the input state is \(\mathinner {|{u}\rangle }\), which is an eigenstate of U with eigenphase \(\phi _u = a_u\sqrt{2} - b_u\), then the history state will contain \(a_u\) terms with one spin in the state \(\mathinner {|{\Omega _a}\rangle }\) and \(b_u\) terms with one spin in the state \(\mathinner {|{\Omega _b}\rangle }\) (each term in the history state will have amplitude \(\frac{1}{T}\)). If the input is a general state \(\mathinner {|{\psi }\rangle }= \sum _u m_u \mathinner {|{u}\rangle }\), then for each u the history state will contain \(a_u\) terms with one spin in the state \(\mathinner {|{\Omega _a}\rangle }\) and \(b_u\) terms with one spin in the state \(\mathinner {|{\Omega _b}\rangle }\), where now each of these terms has amplitude \(m_u / T\).
Let \(\Pi {:}{=}\sum _i \mathinner {|{\Phi (i)}\rangle }\mathinner {\langle {\Phi (i)}|}\) for some complete basis \(\mathinner {|{i}\rangle }\), and we define \(H_1{:}{=}T(\sqrt{2} \Pi _a - \Pi _b)\), where T is the total time in the computation. It thus follows that the energy of \(\mathinner {|{\Phi (u)}\rangle }\) with respect to the operator \(\Pi H_1 \Pi \) is given by \(\phi _u + {{\,\mathrm{O}\,}}(\epsilon )\).
Finally, we need the following technical lemma from [27].
Lemma 4.4
(First-order simulation [27]). Let \(H_0\) and \(H_1\) be Hamiltonians acting on the same space and \(\Pi \) be the projector onto the ground space of \(H_0\). Suppose that \(H_0\) has eigenvalue 0 on \(\Pi \) and the next smallest eigenvalue is at least 1. Let V be an isometry such that \(VV^{\dagger }=\Pi \) and
$$\begin{aligned} \Vert V H_{\mathrm {target}}V^\dag - \Pi H_1 \Pi \Vert \le \epsilon /2. \end{aligned}$$
(16)
Let \(H_{{\text {sim}}} = \Delta H_0 + H_1\) . Then there exists an isometry \({\tilde{V}}\) onto the space spanned by the eigenvectors of \(H_{{\text {sim}}}\) with eigenvalue less than \(\Delta /2\) such that
-
1.
\(\Vert V-{\tilde{V}}\Vert \le {{\,\mathrm{O}\,}}(\Vert H_1\Vert /\Delta )\)
-
2.
\(\Vert {\tilde{V}}H_{{\text {target}}} {\tilde{V}}^{\dagger } -H_{{\text {sim}}< \Delta /2} \Vert \le \epsilon /2 + {{\,\mathrm{O}\,}}(\Vert H_1\Vert ^2/\Delta )\)
We will apply Lemma 4.4 with \(H_0=2T^2H_{\mathrm {SF}}\) and \(H_1=T(\sqrt{2} \Pi _a - \Pi _b)\). We have \(\lambda _{\min }( H_{\mathrm {SF}}) = 0\) and the next smallest nonzero eigenvalue of \(H_{\mathrm {SF}}\) is \((1-\cos (\pi /2T)\ge 1/2T^2)\) by Lemma 4.1, so \(H_0=2T^2H_{\mathrm {SF}}\) has next smallest nonzero eigenvalue at least 1. Moreover, \(\left\| H_1\right\| = \sqrt{2}T\). Note that \(V'\), as defined in Eq. (14), is an isometry which maps onto the ground state of \(H_0\). By construction, we have that the spectrum of \(H_{\mathrm {target}}\) is approximated to within \(\epsilon \) by \(H_1\) restricted to the ground space of \(H_{\mathrm {SF}}\); thus, \(\Vert \Pi H_1 \Pi - \tilde{{\mathcal {E}}}(H)\Vert \le \epsilon \).
Lemma 4.4 therefore implies that there exists an isometry \({\tilde{V}}\) that maps exactly onto the low energy space of \(H_{\mathrm {univ}}\) such that \(\Vert {\tilde{V}}-V'\Vert \le {{\,\mathrm{O}\,}}(\sqrt{2}T/(\Delta /2T^2))={{\,\mathrm{O}\,}}(T^3/\Delta )\). By the triangle inequality and Eq. (15), we have:
$$\begin{aligned} \Vert V-{\tilde{V}}\Vert \le \Vert V-V'\Vert +\Vert V'-{\tilde{V}}\Vert \le O \left( \frac{T^3}{\Delta } + \frac{T_\mathrm {PE}}{T}\right) . \end{aligned}$$
(17)
The second part of the lemma implies that
$$\begin{aligned} \Vert {\tilde{V}} H_{\mathrm {target}}{\tilde{V}}^{\dagger } -H_{{\text {univ}} <\Delta '/2}\Vert \le \epsilon /2+ {{\,\mathrm{O}\,}}((\sqrt{2}T)^2/(\Delta /2T^2))=\epsilon /2 +{{\,\mathrm{O}\,}}(T^4/\Delta ).\nonumber \\ \end{aligned}$$
(18)
Therefore, the conditions of Definition 2.3 are satisfied for a \((\Delta ',\eta ,\epsilon ')\)-simulation of \(H_{\mathrm {target}}\), with \(\eta = O \left( T^3/\Delta + T_\mathrm {PE}/T\right) \), \(\epsilon ' = \epsilon +{{\,\mathrm{O}\,}}(T^4 / \Delta )\) and \(\Delta '= \Delta /2T^2\). Therefore, we must increase L so that \(T \ge {{\,\mathrm{O}\,}}(T_\mathrm {PE}/\eta )={{\,\mathrm{poly}\,}}(n, d^k, \Vert H\Vert ,1/\epsilon ,1/\eta )\) by Eq. (10) (thereby determining x), and increase \(\Delta \) so that
$$\begin{aligned} \Delta \ge \Delta ' T^2 +\frac{T^3}{\eta }+\frac{T^4}{\epsilon } \end{aligned}$$
(19)
to obtain a \((\Delta ', \eta , \epsilon )\)-simulation of the target Hamiltonian. The claim follows.
\(\square \)
We can now prove our main theorem:
Theorem 4.5
There exists a two-body interaction depending on a single parameter \(h(\phi )\) such that the family of translationally invariant Hamiltonians on a chain of length N,
$$\begin{aligned} H_{\mathrm {univ}}(\phi , \Delta , T) = \Delta \sum _{\langle i,j \rangle } h(\phi )_{i,j} + T \sum _{i=0}^{N-1} \left( \sqrt{2} \Pi _{\alpha } - \Pi _{\beta } \right) _i, \end{aligned}$$
(20)
is a universal model, where \(\Delta \), T and \(\phi \) are parameters of the Hamiltonian, and the first sum is over adjacent site along the chain. Furthermore, the universal model is efficient in terms of the number of spins in the simulator system.
Proof
The two-body interaction \(h(\phi )\) makes up a standard form Hamiltonian which encodes a QTM, \(M_1\) dovetailed with the phase-estimation computation from Lemma 4.3. The QTM \(M_1\) carries out phase estimation on the parameter \(\phi \) in the Hamiltonian, and writes out the binary expansion of \(\phi \) (which contains a description of the Hamiltonian to be simulated) on its work tape. There is a standard form Hamiltonian in [14] which encodes this QTM, so by Lemma 4.3 we can construct a standard form Hamiltonian which simulates all quantum Hamiltonians by dovetailing \(M_1\) with \(M_\mathrm {PE}\).
The space requirement for the computation is \({{\,\mathrm{O}\,}}(|\phi |)\), where \(|\phi |\) denotes the length of the binary expansion of \(\phi \), and the computation requires time \(T_1 = {{\,\mathrm{O}\,}}(|\phi |2^{|\phi |})\) [21, Theorem 10] As we commented in Sect. 3.3.1, the standard form clock construction set out in [21, Sect. 4.5] allows for computation time of \( {{\,\mathrm{O}\,}}(|\phi |2^{|\phi |})\) using a Hamiltonian on \(|\phi |\) spins. We therefore find that for a k-local target Hamiltonian \(H_{\mathrm {target}}\) acting on n spins of local dimension d, the number of spins required in the simulator system for a simulation that is \(\epsilon \) close to \(H_{\mathrm {target}}\) is given by \(N = {{\,\mathrm{O}\,}}(|\phi |) ={{\,\mathrm{poly}\,}}\left( n,d^k,\Vert H\Vert ,1/\eta ,1/\epsilon \right) \).
Therefore, the universal model is efficient in terms of the number of spins in the simulator system as defined in Definition 2.4. \(\square \)
Note that this universal model is not efficient in terms of the norm \(\Vert H_{\mathrm {univ}}\Vert \). This is immediately obvious, since \(\Vert H_{\mathrm {univ}}\Vert = \Omega (\Delta )\), and using the relations between \(\Delta '\), \(\eta \), \(\epsilon \), and T and \(\Delta \) from Lemma 4.3 and Eq. (19),
$$\begin{aligned} T= & {} T_1+L+T_\mathrm {PE}\\= & {} O\left( 2^x+{{\,\mathrm{poly}\,}}\left( n,d^k,\Vert H_{\mathrm {target}}\Vert , \frac{1}{\epsilon }, \frac{1}{\eta }\right) \right) \quad \text { and } \quad \Delta \ge \Delta ' T^2 +\frac{T^3}{\eta }+\frac{T^4}{\epsilon } \end{aligned}$$
by Eq. (10), so \(T,\Delta \) are both \({{\,\mathrm{poly}\,}}\left( 2^x,\Vert H_{\mathrm {target}}\Vert ,\Delta ',1/\epsilon , 1/\eta \right) \). For a k-local Hamiltonian \(H_{\mathrm {target}}\) with description x as presented in Sect. 3.2, \(|x|=\Omega \left( md^{2k} \log (\Vert H_{\mathrm {target}}\Vert md^{2k}/\delta )\right) \).
However, if we only wish to simulate a translationally invariant k-local Hamiltonian \(H_{\mathrm {target}}\), this can be specified to accuracy \(\delta \) with just \(\log (\Vert H_{\mathrm {target}}\Vert m d^{2k} /\delta )\) bits of information. In this case (for \(d,k={{\,\mathrm{O}\,}}(1)\) and taking \(\delta =\epsilon \)), the interaction strengths are then \({{\,\mathrm{poly}\,}}(n,\Vert H_{\mathrm {target}}\Vert ,\Delta ', \frac{1}{\eta },\frac{1}{\epsilon })\), and the whole simulation is efficient.
Lemma 4.3 also allows the construction of a universal quantum simulator with two free parameters.
Theorem 4.6
There exists a fixed two-body interaction h such that the family of translationally invariant Hamiltonians on a chain of length N,
$$\begin{aligned} H_{\mathrm {univ}}(\Delta , T) = \Delta \sum _{\langle i,j \rangle } h_{i,j} + T \sum _{i=0}^{N-1} \left( \sqrt{2} \Pi _{\alpha } - \Pi _{\beta } \right) _i, \end{aligned}$$
(21)
is a universal model, where \(\Delta \) and T are parameters of the Hamiltonian, and the first sum is over adjacent sites along the chain.
Proof
As in Theorem 4.5, the two-body interaction h makes up a standard form Hamiltonian which encodes a QTM \(M_1\) dovetailed with the phase-estimation computation from Lemma 4.3. It is based on the construction from [15].
Take \(M_1\) to be a binary counter Turing machine which writes out N—the length of the qudit chain—on its work tape. We will choose N to contain a description of the Hamiltonian to be simulated, as per Sect. 3.2. There is a standard form Hamiltonian in [15] which encodes this QTM, so by Lemma 4.3 we can construct a standard form Hamiltonian which simulates all quantum Hamiltonians by dovetailing \(M_1\) with \(M_\mathrm {PE}\).
Since B(N), as defined in Eq. (7), contains a description of the Hamiltonian to be simulated, we have that
$$\begin{aligned} N = {{\,\mathrm{poly}\,}}\left( 2^{{{\,\mathrm{poly}\,}}(n,\Vert H_{\mathrm {target}}\Vert ,1/\eta ,1/\epsilon )} \right) . \end{aligned}$$
The standard form clock used in the construction allows for computation time polynomial in the length of the chain, so \(\exp ({{\,\mathrm{poly}\,}})\)-time in the size of the target system. As before, by Eq. (10), we require
$$\begin{aligned} T= & {} T_1+L+T_\mathrm {PE}\\= & {} O\left( N+{{\,\mathrm{poly}\,}}\left( n,d^k,\Vert H_{\mathrm {target}}\Vert , \frac{1}{\epsilon }, \frac{1}{\eta }\right) \right) \quad \text { and } \quad \Delta \ge \Delta ' T^2 +\frac{T^3}{\eta }+\frac{T^4}{\epsilon }. \end{aligned}$$
\(\square \)
According to the requirements of Definition 2.3, the universal simulator of the second theorem is not efficient in either the number of spins, nor in the norm. However—as was noted in [2]—this is unavoidable if there is no free parameter in the universal Hamiltonian which encodes the description of the target Hamiltonian: a translationally invariant Hamiltonian on N spins can be described using only \({{\,\mathrm{O}\,}}({{\,\mathrm{poly}\,}}\log (N))\) bits of information, whereas a k-local Hamiltonian which breaks translational invariance in general requires \({{\,\mathrm{poly}\,}}(N)\) bits of information. So, by a simple counting argument, we can see that it is not possible to encode all the information about a k-local Hamiltonian on n spins in a fixed translationally invariant Hamiltonian acting on \({{\,\mathrm{poly}\,}}(n)\) spins.
We observe that the parameters \(\Delta \) and T are qualitatively different to \(\phi \), in that they do not depend on the Hamiltonian to be simulated, but only the parameters \((\Delta ',\epsilon ,\eta )\) determining the precision of the simulation.
No-Go for Parameterless Universality
Is an explicit \(\Delta \)-dependence of a simulator Hamiltonian \(H_{\mathrm {univ}}\) necessary to construct a universal model? Note that an implicit dependence of \(H_{\mathrm {univ}}\) on \(\Delta \) is possible via the chain length \(N=N(\Delta )\) in Theorem 4.5. In the following, we prove that such an implicit dependence is insufficient, by giving a concrete counterexample for which an explicit \(\Delta \)-dependence is necessary.
To this end, we note that it has previously been shown [28] that a degree-reducing Hamiltonian simulation (in a weaker sense of simulation, namely gap-simulation where only the ground state(s) and spectral gap are to be maintained) is only possible if the norm of the local terms is allowed to grow. In order to construct a concrete example in which an explicit \(\Delta \)-dependence is necessary, we first quote Aharonov and Zhou’s result and then translate the terminology to our setting.
Theorem 4.7
(Aharonov and Zhou ( [28, Thm. 1])). For sufficiently small constants \(\epsilon \ge 0\) and \({\tilde{\omega }}\ge 0\), there exists a minimum system size \(N_0\) such that for all \(N\ge N_0\) there exists no constant-local \([r,M,J]=[{{\,\mathrm{O}\,}}(1),M,{{\,\mathrm{O}\,}}(1)]\) gap simulation (where r is the interaction degree, M the number of local terms, and J the local interaction strength of the simulator) of the Hamiltonian
with a localized encoding, \(\epsilon \)-incoherence, and energy spread \({\tilde{\omega }}\), for any number of Hamiltonian terms M.
Corollary 4.8
Consider a universal family of Hamiltonians with local interactions and bounded-degree interaction graph. Hamiltonians in this family must have an explicit dependence on the energy cut-off (\(\Delta \)) below which they are valid simulations of particular target Hamiltonians.
Proof
We first explain the notation used in Theorem 4.7. As mentioned, the notion of gap simulation is weaker than Definition 2.3. Only the (quasi-) ground space \({\mathcal {L}}\) of \(H_A\), rather than the full Hilbert space, needs to be represented \(\epsilon \)-coherently: \(\Vert H_A|_{{\mathcal {L}}} - {{\tilde{H}}}_A|_{{\mathcal {L}}}\Vert < \epsilon \), where \(\cdot |_{{\mathcal {L}}}\) denotes the restriction to \({\mathcal {L}}\)). And only the spectral gap above the ground space, rather than the full spectrum, must be maintained: \({\tilde{\gamma }}=\Delta ({{\tilde{H}}}_A) \ge \gamma = \Delta (H_A)\). The rest of the spectrum in the simulation can be arbitrary. Energy spread in this context simply means the range of eigenvalues within \({\mathcal {L}}\) spreads out at most such that \(|\lambda _0 - {\tilde{\lambda }}_0|\le {{\tilde{\omega }}}\gamma \).
A \([{{\,\mathrm{O}\,}}(1),M,{{\,\mathrm{O}\,}}(1)]\) simulation with the above parameters then simply means an \(\epsilon \)-coherent gap simulation, constant degree and local interaction strength, where M—the number of local terms in the simulator—is left unconstrained, and the eigenvalues vary by at most \({\tilde{\omega }}\gamma \).
It is clear that this notion of simulation falls within our more generic framework of simulation (cf. [28, Sec. 1.1]): a simulation of \(H_A\) also defines a valid gap simulation of \(H_A\). Since by Definition 2.4 this simulation can be made arbitrarily precise, with parameters \(\epsilon ,{{\tilde{\omega }}}\) arbitrarily small, and has constant interaction degree by assumption, this contradicts Theorem 4.7. \(\square \)