Observation of a non-Hermitian supersonic mode on a trapped-ion quantum computer (2025)

This section details the numerical and analytical methods used throughout the work. We give a motivating example of non-Hermitian physics, followed by the discussion of sequential circuits generated by Matrix Product States (MPS) and Gaussian Matrix Product States (GMPS), explaining how they enable efficient quantum representations of near-area-law quantum states. Additionally, we provide an analytical proof of Theorem I.1.

Non-Hermitian physics: a motivating example

We argued in the main text that a physical way to motivate non-Hermitian physics is from open quantum system dynamics. Usually, the Linbladian equation is used to describe the Markovian evolution of a system interacting with a thermal bath⁵³:

Here, the system density matrix ρ evolves under H, the system Hamiltonian, and {L_i} is a set of jump operators corresponding to the dissipations due to the bath. In the second line, we have redefined \({H}_{{{\rm{eff}}}}=H-\frac{i}{2}{\sum}_{i}{\gamma }_{i}{L}^{{\dagger} }L\).

The last term in Eq. (13), \({\sum}_{i}{\gamma }_{i}{L}_{i}\rho {L}_{i}^{{\dagger} }\), describes quantum jump processes that can be associated with a measurable physical quantity, such as a spontaneous emission of a photon. The stochastic time evolution can be classified into different quantum trajectories depending on the number of quantum jumps. Now, if we postselect on the absence of quantum jumps, we end up with a Schrodinger-like time evolution with a non-Hermitian Hamiltonian:

Review of sequential circuits

In a quantum system consisting of n qubits, the sequential quantum circuits are defined as follows:

Definition 4.1

(τ-sequential quantum circuit) Given a local universal gate set \({{\mathcal{G}}}\subseteq U(4)\), a quantum circuit consisting of local gates in the set is called a τ-sequential quantum circuit if each qubit is at most acted upon by τ gates in the circuit.

Various interesting examples emerge at τ ≪ n. For example, when τ is a constant, from the definition, we can bound its computational power:

1. 1.

   Strictly contains all constant depth circuits;

2. 2.

   Is strictly contained in the set of linear depth circuits.

The first inclusion holds because any constant-depth circuit is a sequential circuit and sequential circuits can prepare long-range correlates that cannot be accessed from constant-depth circuits such as the GHZ state. The second bound can be deduced from a) these sequential circuits are at most depth O(n), as their circuit volume is O(n) by definition and b) they cannot generate volume law entanglement states, which are permitted in linear depth circuits.

As such, sequential circuits are of particular interest in the NISQ era: Compared with a constant depth circuit, it has more representation power as its light cone can cover the whole system; compared to a ‘dense’ constant depth circuit, it permits a compressed representation yet accurately captures certain classes of states, such as thermal states of locally interacting spin chains⁵⁴, electronic mean-field ground states⁴⁰, chiral topological orders⁵⁵, maps between gapped phases⁵⁶, projected entangled-pair states⁵⁷, etc.; they can be combined with adaptive circuits to improve their representation power^58,59,60, and experimental proposals on cQED devices have been proposed^61,62.

Matrix product states

The discussion of sequential circuits originated from a celebrated quantum state compression method: MPS. Any pure quantum state \(\left\vert \Psi \right\rangle\) can be expressed as a matrix-product state by sequentially performing Schmidt decompositions between local sites, turning wave-function amplitudes into a 1D tensor train (in physical states considered in this work, we focus on open boundary 1D chains with sites x ∈ {1, 2, …n}, although the method can be generalized to infinite systems and larger on-site dimensions):

In this context, s_x ∈ {0, 1} labels the basis states for site x, and \({A}^{{s}_{x}}\) are χ×χ matrices. The vectors ℓ are χ-dimensional and determine the left boundary conditions. The memory and computational cost of Matrix Product State (MPS) computations scale with the bond dimension, χ, which is lower bounded by the bipartite entanglement entropy across a cut through the bond.

Although representing a generic quantum state, such as the Haar random state still requires the bond dimension χ=e^O(n), many quantum states of physical interests such as 1D short-range correlated, area-law entangled states^40,52,63, or thermal mixed states^54,64, it is possible to truncate the entanglement spectrum to χ=O(1) independent of system size, allowing for efficient classical simulations of 1D gapped ground states.

Higher-dimensional systems can also be represented as MPS by treating them as a 1D stack of (d−1)-dimensional cross-sections. Yet even for area-law entangled states the required bond dimension grows exponentially with the cross-sectional area, making classical simulation impractical. In such cases including classically intractable cases such as 2D and 3D ground-states with symmetry-breaking⁴⁰ or (non-chiral) topological order⁶⁵, and finite-time quantum dynamics from any qMPS^31,66, applying tensor network methods on a quantum computer could offer a significant advantage.

Sequential quantum circuits generated by MPS (qMPS)

Properties of any MPS in right-canonical form (RCF)³² can be measured by sampling on a quantum computer and implementing its transfer-matrix as a quantum channel⁶⁷ acting on 1 “physical” qubit and \(q={\log }_{2}\chi\) bond qubits (see Fig.2 for a graphical representation). Each tensor A is then embedded as a block of larger unitary operator U_A acting on a fixed initial state \(\left\vert 0\right\rangle\) of the physical qubits. After the application of U_A, the physical qubits can be measured in any desired basis while entanglement information is consistently stored on the bond qubits. The process is then repeated for each site in sequence from left to right. In this way, one can measure any product operator of the form \({\prod }_{x=1}^{n}{{{\mathcal{O}}}}_{x}\), which forms a complete basis for general observables. Crucially, once measured, the physical qubit for site x can be reset to \(\left\vert 0\right\rangle\) and reused as the physical qubits for site x+1, enabling a small quantum processor to achieve quantum simulation tasks with sizes far larger than the number of qubits available³¹.

To summarize, the qMPS procedure for sampling an observable of the form \(\left\langle \psi \right\vert \mathop{\prod }_{x=1}^{n}{O}_{x}\left\vert \psi \right\rangle\) is:

Algorithm 3

– Generating sequential quantum circuits from MPS

1: Prepare the bond qubits in a state corresponding to the left boundary vector ℓ. This can be done with up to \(\log \chi\) ancilla qubits

2: for x=1…n do

3: Perform a synthesized quantum circuit U_A at site x, entangling the physical and bond qubits.

4: Measure the physical qubit in the eigenbasis of O_x and weight the measurement outcome by the corresponding eigenvalue of that observable.

5: Reset the physical qubit for site x in a fixed reference state, \(\left\vert 0\right\rangle\).

6: end for

7: Discard the bond qubits.

Moreover, the entanglement spectrum of the bond-qubits in between sites x and x+1 coincides with the bipartite entanglement spectrum of the physical MPS at that entanglement cut, further enabling measurement of non-local entanglement observables, as recently demonstrated experimentally³¹. The left boundary vector ℓ is prepared by a unitary circuit acting on the bond space. For an open chain, there is no need to specify the right boundary condition as no entanglement is to be stored beyond x=n, and thus the bond qubits are disentangled with the physical qubit and can be traced out. When all U_A matrices are set to be the same, the lack of right boundary conditions in the formalism describes a semi-infinite wire³¹.

By exploiting the efficient compression⁶⁸ of physically interesting states, such as low-energy states of local Hamiltonians, qTNS methods enable simulation of many-body systems relevant to condensed-matter physics and materials science with much smaller quantum memory than would be required to directly encode the many-body wave-function.

Gaussian MPS

While MPS is a generic approach to quantum state compression, a subclass of MPS, the Gaussian MPS (GMPS), explores the near-area law entanglement of free fermion systems, enabling even more efficient representations. The Hamiltonian of a Gaussian (i.e. non-interacting) fermion system with n sites has the form

This system can be fully characterized by its n×n two-point Green’s function \({G}_{ij}=\langle {c}_{i}^{{\dagger} }{c}_{j}\rangle\) with highly degenerate eigenvalues of either 0 (unoccupied) or 1 (occupied sites). Crucially, G_ij remains invariant under any unitary transformation as long as one does not mix occupied and unoccupied states.

The compression scheme presented by Fishman and White³⁹ exploits this unitary invariance by progressively disentangling local degrees of freedom in blocks of B adjacent sites. Moreover, the ground states of Gaussian fermionic systems have near-area-law entanglement. Therefore, choosing block size B large enough, most of the block eigenvalues must be exponentially close to 0 or 1. This enables a sequence of operations that compresses the correlation matrix:

Algorithm 4

– GMPS compression

1: for x=1…n do

2: Start with G_xx, examine the next B×B sub-matrix of G to the bottom right.

3: Identify the eigenvector with an eigenvalue closest to 0 or 1.

4: Apply a series of 2×2 single-particle unitary rotations \({\prod }_{\alpha=1}^{B-1}{u}_{x,\alpha }^{{\dagger} }\) on G to move this eigenvector to the first site of the block. Denote the resultant correlation matrix as \({G}^{{\prime} }\).

5: Set \(G\leftarrow {G}^{{\prime} }\)

6: end for

In each iteration, the compression algorithm separates a site from the rest of the system. At the end of execution, Green’s function is approximately diagonalized. Reversing the process allows one to transform a product state into the desired Gaussian fermionic state.

GMPS as a sequential quantum circuit

To simulate fermions on a quantum computer, these single-particle operations u_α in the fermionic language, can be converted into a circuit for the many-particle Hilbert space of size 2ⁿ by replacing each 2×2 unitary \({u}_{x,\alpha }^{{\dagger} }\) with a two-qubit gate:

As pictured in Fig.1(b), the resulting ladder circuit U=∏_x,α U_x,α can be interpreted as a B-sequential quantum circuit generated by MPS (Fig.2) with bond dimension χ=2^B whose causal cone slightly differentiates from generic non-Gaussian (q)MPS of the same bond dimension. In the 1D Hamiltonian considered in Eq. 5 and J_z=0, we numerically find that it suffices to choose block size B=3 to prepare the ground state to energy infidelity <1% on a n=18 chain.

Directly preparing an arbitrary Gaussian state with a ladder circuit on n qubits requires O(n²) two-qubit gates (as shown in⁶⁹). To compare, a compressed GMPS ground state can be prepared with O(nB) two-qubit gates acting on O(B) qubits when implemented sequentially. The efficiency of this compression depends on the block size B needed for accurate state approximation.

Numerical evidence and entanglement-based arguments suggest that for ground states of local Hamiltonians in 1D systems of length L with a target error threshold \(\epsilon=1-\frac{1}{L}{\sum}_{i,j}| {G}_{ij}^{(\,{\mbox{GMPS}})}-{G}_{ij}|\), the required block size B scales as \(\log {\epsilon }^{-1}\) for gapped states or \(\log L\log {\epsilon }^{-1}\) for gapless metallic states. In Niu et al.⁴⁰, these results are extended to d-dimensional systems, where B generally scales with the bipartite entanglement entropy S(L):

This result holds even for topologically non-trivial Chern band insulators that have an obstruction to forming a fully localized Wannier-basis. Compared to standard adiabatic state-preparation protocols, this method dramatically reduces the number of qubits required (L^d−1B vs. L^d) to implement the GMPS on a quantum computer⁴⁰; the compressed state can be then used as an initial state for quantum quench or adiabatic state preparations.

Analytical proof of Thm. I.1 and implication in complexity theory

In this section, we provide analytical proof of Theorem I.1, which says there exists a non-Hermitian H consisting of single qubit terms only and an initial state \(\left\vert {\psi }_{0}\right\rangle\) where the dynamics for \(\Theta (\log (n))\) time can be exponentially hard to approximate with a quantum circuit. One explicit example is to take \(\left\vert {\psi }_{0}\right\rangle\) to be a Haar random state \(\left\vert {\psi }_{{{\rm{Haar}}}}\right\rangle\) and \({H}_{z}=-i{\sum}_{i}{Z}_{i}\). Our hardness of approximation result comes from two facts:

1. 1.

   Applying the dynamics generated by H_z for evolution time \(t=\Theta (\log (n))\) and measure in the computational basis allows one to distinguish a Haar random state from a maximally mixed state \({\rho }_{m}={\mathbb{I}}/d\), where d=2ⁿ.

2. 2.

   With overwhelmingly high probability, distinguishing a state sampled from Haar random ensemble from ρ_m is exponentially hard.

The task can be formalized into a decision problem:

Problem 4.2

(Distinguishes a Haar random state from a maximally mixed state) An oracle \({{\mathcal{O}}}\) prepares copies of a fixed quantum state on a n-qubit quantum register that is promised to be either a maximally mixed state or a pure state sampled Haar randomly. Decide which is the case with as few queries as possible.

We first prove that time evolution with a single qubit dissipation channel can efficiently solve Problem IV.2.

Lemma 4.3

(Single qubit dissipation distinguishes a Haar random state from a maximally mixed state) With O(k) queries and probably 1−e^−O(k), a quantum state sampled Haar randomly can be distinguished from a maximally mixed state by time evolving the state for \(t=\Theta (\log (n))\) with H_z.

Proof

We consider the output distributions of a Haar random state before and after time evolution, measured on the computational basis

p_s is known to fluctuate around its mean value i.e. the distribution of p_s=1/d, but with a variance exponentially small in n. This can be seen from the numerical experiments on the left panel of Fig.4. As a result, it is exponentially hard to distinguish a Haar random state and a maximally mixed state with a naive computational basis measurement.

The effect of the time evolution generated by H_z is that it exponentially re-distributes weights of all output strings according to their Hamming weight w_s, or the number of 0’s in s. This can already be told from a single qubit case:

We begin by finding t, such that after applying the evolution generated by H_z to the maximally mixed state, the output weight on the s=1ⁿ can be a constant c. By setting:

and solving for t, we get

What would the output distribution look like for a Haar random state after applying the same time evolution? The normalized weight is just

In the second from last step, we use the fact that each entry of a Haar state vector consists of two standard normal variables and the weights on each string should follow the Porter-Thomas distribution: \({{\rm{PT}}}({p}_{s})={2}^{n}{e}^{-{2}^{n}{p}_{s}}\), and a sum over poly(n) terms would converge to its mean value. We may also ignore the \({p}_{{0}^{n}}{e}^{-2n}\) in the denominator as it is exponentially small in n.

Under the assumption of Eq. (20), we have

W.l.o.g., setting c=1/2 gives η=1; therefore the Eq. (28) returns

This means the variance of this distribution is now a constant independent of n. For example, the probability of \(P[{q}_{{1}^{n}}\ge 0.6]\) is

As evident from Fig.4, with constant probability, a measurement in the computational basis now allows efficient distinguishing between the Haar random state and the maximally mixed state after the time evolution. Notice that, if we choose t=poly(n), the two states are once again hard to be distinguished because they both purify to a product state. This probability can be boosted to 1−e^−k by randomly applying a layer of X gates before the time evolution (namely, randomly selecting a string s to amplify and probe its amplitude) and repeating O(k) times. □

In the unitary setting, the minimum number of gates required to implement a measurement M that can distinguish a given quantum state \(\left\vert \psi \right\rangle\) from the maximally mixed state ρ_m to a certain resolution δ is defined as the strong state complexity. Formally, let \(\beta (r,\left\vert \psi \right\rangle )\) be the maximum bias with which \(\left\vert \psi \right\rangle\) can be distinguished from the maximally mixed state using a circuit with at most r gates from the gate set \({{\mathcal{G}}}\subseteq U(4)\):

Then the strong state complexity is defined as:

Definition 4.4

(Strong state complexity) For a given \(r\in {\mathbb{N}}\) and η ∈ (0, 1), a pure state \(\left\vert \psi \right\rangle\) has strong η-state complexity at most r if and only if \(\beta (r,\left\vert \psi \right\rangle )\ge 1-1/d-\delta\). We denote this \({{{\mathcal{C}}}}_{\delta }(\left\vert \psi \right\rangle )\le r\).

The 1/d in the definition comes from that any pure state can be trivially distinguished from a maximally mixed state with trace distance 1/d. For Haar random pure states, it has been shown that the vast majority of them have exponentially high strong state complexity:

Proposition 4.5

(Strong state complexity of Haar random states) The probability that \(\left\vert {\psi }_{{{\rm{Haar}}}}\right\rangle\) has strong circuit complexity less than r is

The proof given by⁴³ essentially comes from a counting argument that any fixed measurement can only distinguish a small number of states, and thus to distinguish the vast majority of Haar random states, the complexity of measurement must grow exponential in n. Even for 1−δ²=1/poly(n), this probability of distinguish remains exponentially small in n when r=poly(n). This means

Lemma 4.6

(Hardness of distinguishing a Haar random state with a BQP machine) With overwhelming probability, a BQP machine making polynomial queries to \({{\mathcal{O}}}\) cannot solve Problem IV.2.

Combining Lemma IV.3 and Lemma IV.6 completes the proof of Thm. I.1. Further, as we explain in Supplementary Note1, each single-qubit dissipation channel can be implemented with a two-qubit gate and post-selection on the single ancillary qubit. It turns out that the result we have proved can be summarized in a computational complexity language: With overwhelming probability, Problem IV.2 can be solved by a quantum circuit at merely constant depth when postselection is allowed (denoted as PostQNC₀), but also with overwhelming probability, it cannot be solved by a polynomial-size quantum circuit. Namely,

Corollary 4.7

(Oracle separation) There exists a quantum oracle \({{\mathcal{O}}}\) (as defined in Problem IV.2) such that \({{{\rm{PostQNC}}}}_{0}^{{{\mathcal{O}}}} \nsubseteq {{{\rm{BQP}}}}^{{{\mathcal{O}}}}\); relative to the same oracle, \({{{\rm{BQP}}}}^{{{\mathcal{O}}}}\subset {{{\rm{PostBQP}}}}^{{{\mathcal{O}}}}\).