Qubit vitrification and entanglement criticality on a quantum simulator

ABSTRACT Many elusive quantum phenomena emerge from a quantum system interacting with its classical environment. Quantum simulators enable us to program this interaction by using measurement

operations. Measurements generally remove part of the entanglement built between the qubits in a simulator. While in simple cases entanglement may disappear at a constant rate as we measure

qubits one by one, the evolution of entanglement under measurements for a given class of quantum states is generally unknown. We show that consecutive measurements of qubits in a simulator

can lead to criticality, separating two phases of entanglement. Using up to 48 qubits, we prepare an entangled superposition of ground states to a classical spin model. Progressively

measuring the qubits drives the simulator through an observable vitrification point and into a spin glass phase of entanglement. Our findings suggest coupling to a classical environment may

drive critical phenomena in more general quantum states. SIMILAR CONTENT BEING VIEWED BY OTHERS BENCHMARKING HIGHLY ENTANGLED STATES ON A 60-ATOM ANALOGUE QUANTUM SIMULATOR Article Open

access 20 March 2024 THE RANDOMIZED MEASUREMENT TOOLBOX Article 02 December 2022 MEASUREMENT-INDUCED TOPOLOGICAL ENTANGLEMENT TRANSITIONS IN SYMMETRIC RANDOM QUANTUM CIRCUITS Article 04

January 2021 INTRODUCTION The Born rule, which states that the outcome of a measurement performed on a quantum state is a random variable whose probability distribution is determined by

quantum theory, governs information transfer from a quantum system to its classical environment. While the Born rule is at play constantly all around us since all matter is fundamentally

quantum, its effects are only evidenced in bulk due to the astronomical number of measurement events that occur at macroscopic lengths and time scales. In contrast, a quantum simulator, a

programmable array of qubits, is an otherwise isolated quantum system that we can couple to its environment at will with tailor-made measurements of all or some of the qubits. As such, it

allows us to study in detail how the quantum characteristics of a system change as we progressively measure its components. If we were to pick the state of an ideal quantum simulator

uniformly at random from the set of all states accessible to it, we would get a _volume-law state_: a state whose _entropy of entanglement_ of an extensive subsystem, measured in bits, is

proportional to the number of qubits. Measuring a single qubit in such a state removes roughly one bit of entanglement. One therefore expects that the entanglement of a volume-law state

should decrease as a simple linear function of the number of measured qubits, yielding a classical unentangled state once we measure all the qubits. However, this is not always the case: the

behaviour of entanglement in a quantum simulator can change dramatically and abruptly as we progressively measure its qubits, exhibiting the phenomenon of _criticality_1. Critical behaviour

indicates a transition between two distinct _phases_ of entanglement. Previous work revealed entanglement phase transitions in different settings, namely, in random ensembles of quantum

circuits in which qubits undergo measurement at a finite rate2,3,4,5,6,7,8 and in models with topological order9,10,11. Detecting and characterizing such critical behaviour in experiments is

challenging though. First, current quantum processors are faulty, limiting the quantum gates we can reliably implement and the number of error-free measurements we can obtain. Second,

measuring the entanglement entropy of an arbitrary quantum state is hard. The straightforward approach requires tomographic reconstruction of the quantum state, which is intractable for

quantum systems with many components. Finally, although theoretical models for entanglement phase transitions have been introduced in the context of random circuit ensembles2,3,5,8, these

are only approximate in the experimentally relevant limit. Here, we program two entanglement phases and the criticality between them on a quantum simulator of up to 48 superconducting

qubits. We do this by implementing an ensemble of quantum circuits1 that allow us to reliably generate volume-law states whose entanglement we can deliberately decrease with qubit

measurements, experimentally determine the entanglement entropy, and capture the exact dependence of entanglement on measurements using a physical theory1 (see “Measuring entanglement”). The

pertinent physical theory is that of spin _vitrification_, i.e., the transition to a spin glass phase12,13. We detect the vitrification point, which agrees with spin glass theory. Our work

shows measurements alone can trigger entanglement criticality, suggesting a classical environment could induce critical behaviour in more general quantum states. RESULTS THEORY AND MODEL Our

experimental system is an array of superconducting qubits on a quantum simulator. To drive these qubits through an entanglement phase transition, we first execute quantum circuits that

prepare highly entangled states. The circuits and theory that follow come from previous theoretical work1 on entanglement phase transitions. Each of our circuits implements a system of _R_

linear equations on _L_ Boolean variables using _R_ + _L_ = _N_ qubits. (In practice, we use fewer qubits to implement our system on hardware. See “Circuit optimization” in the “Methods”.)

We can write the system as the matrix equation $$B{{{{{{{\bf{x}}}}}}}}={{{{{{{\bf{y}}}}}}}}\,{{{{{{\mathrm{mod}}}}}}}\,\,2\,,$$ (1) where _B_ is an _R_ × _L_ Boolean matrix whose rows

represent equations and columns represent variables, as shown in Fig. 1a. If _B__i__j_ = 1, then equation _i_ involves variable _j_. Otherwise, the entry is zero. By setting the elements of

_B_ according to some distribution, we get an ensemble of matrices. Each element of the parity vector \({{{{{{{\bf{y}}}}}}}}\in {\left\{0,\, 1\right\}}^{R}\) fixes the parity of an equation

and each \({{{{{{{\bf{x}}}}}}}}\in {\left\{0,\, 1\right\}}^{L}\) is a solution to the system for a given Y. To implement the system of Eq. (1) using a quantum circuit, we organize the qubits

in the simulator in two registers, as sketched in Fig. 1b,c: a “variable” register consisting of _L_ variable qubits and a “parity” register consisting of _R_ parity qubits. We input

variable qubits in the \(|\!+\! \rangle\) state and parity qubits in the \(\left|0\right\rangle\) state into a circuit from the ensemble defined above. The initial state is therefore

\({|\psi \rangle }_{{{{{\rm{in}}}}}}={|\!+\rangle }^{\otimes L}{|0\rangle }^{\otimes R}\). The state \({|\psi \rangle }_{{{{{{{{\rm{out}}}}}}}}}\) at the output of the circuit is an

entangled equal superposition of solutions X for each possible Y (see “Quantum state and entanglement entropy” in the “Methods” for the exact state). For each Y, the set of solutions

\(\left\{{{{{{{{\bf{x}}}}}}}}\right\}\) is unique. Each parity qubit at the output holds the parity of the variables that appear in the corresponding row of _B_ (see Fig. 1b). Because these

variable qubits are in a superposition of classical states, so is the parity qubit: it is 0 or 1 depending on the state of the variable qubits. The variable qubits are thus entangled with

the parity qubit and contribute one bit of entanglement. What is the total entanglement between variable and parity qubits? We quantify this with the entanglement entropy _S_, which counts

the number of bits of entanglement between two parts of a quantum system. To answer our question above, we need to know how many possible vectors Y there are in the superposition. For each

of the rank(_B_) linearly independent rows of _B_, the corresponding component in Y can be set to zero or one freely. There are then _N_Y = 2rank(_B_) possible vectors Y which give solutions

to Eq. (1). This means the entanglement entropy between the variable and parity qubits is \(S({|\psi \rangle }_{{{{{{{{\rm{out}}}}}}}}}) \sim {\log

}_{2}{N}_{{{{{{{{\bf{y}}}}}}}}}={{{{{{{\rm{rank}}}}}}}}(B)\). ENTANGLEMENT PHASE TRANSITION Each time we measure a qubit in a generic volume-law state, the system loses one bit of

entanglement. This happens with every measurement, reducing the number of superposed configurations until just a single classical state remains. The states \({|\psi \rangle

}_{{{{{{{{\rm{out}}}}}}}}}\) behave in a manifestly different manner. To prove this, we start by compiling enough linear equations into our circuits such that an output state is a

superposition of _N_Y = 2rank(_B_) = 2_L_ vectors Y. Then, we calculate the entanglement entropy after measuring a subset _M_ of the parity qubits in our system in the computational basis.

We label the measurement outcome Yout,_M_ and the partially measured state \({\left|\psi \right\rangle }_{{{{{{{{\rm{out}}}}}}}},M}\). The state \({\left|\psi \right\rangle

}_{{{{{{{{\rm{out}}}}}}}},M}\) still contains an equal superposition of solutions to Eq. (1), but the elements of the parity vector Y that correspond to the subset _M_ are fixed to the

measurement outcome Yout,_M_. For the same reasoning as in the previous section, there are \({N}_{{{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M}}={2}^{{{{{{{{\rm{rank}}}}}}}}({B}_{M})}\)

equally probable measurement outcomes Yout,_M_, where _B__M_ are the rows of _B_ that correspond to the _M_ parity qubits. (In Fig. 1c, measuring the first three parity qubits would mean

_B__M_ is the first three rows of _B_.) Since measuring the output state determines Yout,_M_, there are

\({N}_{{{{{{{{\bf{y}}}}}}}}}/{N}_{{{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M}}={2}^{L-{{{{{{{\rm{rank}}}}}}}}({B}_{M})}\) vectors Y remaining in the state \({|\psi \rangle

}_{{{{{{\rm{out}}}}}},M}\). The entanglement entropy is then \(S({|\psi \rangle }_{{{{{{{{\rm{out}}}}}}}},M}) \sim {\log

}_{2}({N}_{{{{{{{{\bf{y}}}}}}}}}/{N}_{{{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M}})=L-{{{{{{{\rm{rank}}}}}}}}({B}_{M})\). Therefore, the evolution of the entanglement entropy is given

by rank(_B__M_) as a function of _M_. To obtain a size-independent control parameter, we define _α_ ≡ ∣_M_∣/_L_, the ratio of measured parity qubits to variable qubits. A unique feature of

states \({\left|\psi \right\rangle }_{{{{{{{{\rm{out}}}}}}}}}\) is that we can obtain an exact result for the evolution of the entanglement entropy under measurements through a mapping to a

classical spin model1. In this description, \({\left|\psi \right\rangle }_{{{{{{{{\rm{out}}}}}}}}}\) is an entangled superposition of ground states X to a spin Hamiltonian with couplings

defined by Y (see “Methods”). We can then use the characteristics of the spin model both to predict the behaviour of the entanglement and, more importantly, to measure it on a quantum

simulator, as we discuss in the next section. To get concrete predictions for our experiments, we now specify the distribution we sample to populate the matrix _B_ and define the ensemble of

states \({\left|\psi \right\rangle }_{{{{{{{{\rm{out}}}}}}}}}\). We pick three distinct variables uniformly at random for each equation in Eq. (1), and we ensure there are no repeated

equations. With this choice, we get an exact correspondence between our output state and the ground states to an instance of the unfrustrated 3-spin model, both of which are given by the

solutions to Eq. (1)14,15. This model exhibits a phase transition at _α__c_ ≈ 0.918. For _α_ < _α__c_, our system corresponds to a paramagnet in the 3-spin model. We thus get a

_paramagnetic_ phase of entanglement. Here, rank(_B__M_) = ∣_M_∣ = _L__α_. This happens because there are few rows in _B__M_, which makes it highly probable that they are linearly

independent. The entanglement entropy of the quantum system after measuring ∣_M_∣ parity qubits scales linearly in both _L_ and _α_: \(S({\left|\psi \right\rangle

}_{{{{{{{{\rm{out}}}}}}}},M}) \sim L\left(1-\alpha \right)\), i.e., the output state obeys a volume law. For _α_ > _α__c_, the system enters a spin glass phase. The qubits vitrify,

turning into an entangled superposition of spin glass ground states. We thus get a _glassy_ phase of entanglement. Now, rank(_B__M_) < ∣_M_∣ because there are many rows in _B__M_ and

linear independence is lost. The entanglement entropy still scales linearly in _L_ but decreases slower than linearly with increasing _α_. We sketch how the measurement of parity qubits

collapses the output state in Fig. 2. The two entanglement phases give rise to different behaviours. In the paramagnetic phase, measuring a parity qubit collapses its state to one of two

equally probable values (Fig. 2b). This occurs when a parity qubit is independent of the other measured parity qubits, which is the case when _B__M_ has full rank. Measuring the parity qubit

halves the number of possible vectors Y remaining in the superposition, so the system loses one bit of entanglement entropy. In the spin glass phase, there is a finite probability that a

parity qubit has a definite value before measurement (Fig. 2c). This occurs when previous measurement outcomes determine the measurement outcome for the next parity qubit, which begins when

rank(_B__M_) < ∣_M_∣. In this case, measuring the parity qubit does not change the number of possible vectors Y, so the entanglement entropy remains the same. MEASURING ENTANGLEMENT The

exact correspondence between spin glass physics and entanglement established above and in ref. [1] gives us an efficient way to detect the entanglement phases and criticality on a quantum

simulator. In our setup, the entanglement entropy is characterized by the spin glass order parameter 16,17,18 $$q({B}_{M})=\frac{1}{L}\mathop{\sum }\limits_{i=1}^{L}{\langle

{(-1)}^{{x}_{i}}\rangle }^{2},$$ (2) where 〈…〉 is an average over all the solutions X for Eq. (1) with matrix _B__M_ and a given parity vector Yout,_M_, and _x__i_ is the _i_-th variable in

X. We derive the identity that links this order parameter to the entanglement entropy in the “Methods”. Therefore, while in most quantum systems quantifying entanglement is intractable, here

we have direct access to the entanglement entropy through the spin glass order parameter, which is efficiently measurable (see “Methods”). The order parameter describes the tendency of the

solutions X to take the same value on each variable. It is zero in the paramagnetic phase, which means a given variable does not have correlations across solutions. The outcome of the

measurement of a parity qubit is independent of previous parity measurements in this phase. At _α_ = _α__c_, the variables abruptly become correlated across solutions, and the order

parameter jumps to a finite value, eventually saturating to one. This implies solutions of the system are almost identical, differing on only a few variables. The outcome of the measurement

of a parity qubit now depends on previous parity measurements. In the language of physics, each variable can be thought of as one of _L_ spins in a many-body system with ∣_M_∣ interactions.

Then, each basis state in \({\left|\psi \right\rangle }_{{{{{{{{\rm{out}}}}}}}},M}\) of the variable qubits represents a spin configuration. In the paramagnetic phase where there are few

interactions, these configurations have no correlation, leading to no order (_q_ = 0). However, in the spin glass phase where ∣_M_∣ > _L__α__c_, the interactions induce correlations

across the configurations, leading to spin glass order (_q_ > 0). Using arrays of up to 48 qubits, our experimental results (Fig. 3) clearly reveal the two entanglement phases and the

transition between them, and are in agreement with theory. The transition at a critical value _α__c_ becomes more abrupt with increasing system size, exactly as spin glass physics dictates.

Finite-size scaling (see “Methods”) of the data using the scaling form \(q(\alpha )=f((\alpha -{\alpha }_{c,\exp }){L}^{1/{\nu }_{\exp }})\) yields the experimental values for the critical

measurement ratio \({\alpha }_{c,\exp }=0.95\pm 0.06\), which agrees with the theoretical value _α__c_ ≈ 0.91815, and the critical exponent \({\nu }_{\exp }=2.5\pm 0.5\). We note that while

there are some similarities between the spin glass order we find and those in other work on entanglement criticality19,20,21, there are a few differences. First, previous work focuses on

states and circuits that respect certain symmetries, which play a role in creating a spin glass phase. In contrast, we do not need to impose symmetries. Second, there is an exact relation

between the spin glass order parameter in our work and the entanglement entropy, which is not there in other works. Third, previous work focuses on spin glass order as a steady state

property of the system, whereas our system goes into a spin glass state immediately after applying our circuit and measuring. Finally, as our results demonstrate, we can observe this spin

glass order on existing quantum hardware. DISCUSSION Since entanglement is a key resource for quantum computation, precise predictions and experimental verification of its possible

behaviours in quantum devices under measurement are sought-after. Our findings demonstrate that partial measurements of quantum states can alone give rise to intricate phenomena related to

entanglement. Measurements can force qubits to vitrify, and hence realize the celebrated13 spin glass phase of matter inside a quantum processor. The spin glass quantum states implemented

here are a subset of stabilizer states, an important class of states for quantum computation. Moreover, we already know that spin glass entanglement criticality is also present in more

general classes of states than the ones studied here1. It is interesting to ask whether similar physics applies to entanglement in monitored quantum systems at large, giving rise to

different types of nonanalyticity for the entanglement entropy. METHODS SPIN HAMILTONIAN The output of our circuits provides X and Y from Eq. (1), which we can relate to the ground states

and couplings of a _p_-spin model22 (with _p_ = 3). The model consists of _L_ spins, with _R_ interactions encoded by the matrix _B_ (see the example in Fig. 1a). The indices of the nonzero

elements in each row _a_ ∈ _B_ correspond to the spins which are part of an interaction. The Hamiltonian is: $$H(B,\, {{{{{{{\boldsymbol{\sigma }}}}}}}},\,

{{{{{{{\bf{J}}}}}}}})=\frac{1}{2}\mathop{\sum}\limits_{a\in B}\left(1-{J}_{a}{\sigma }_{{a}_{1}}{\sigma }_{{a}_{2}}{\sigma }_{{a}_{3}}\right),$$ (3) where _a_1, _a_2, _a_3 refer to three

distinct spins (the indices of the nonzero elements in _a_), _J__a_ are the couplings of the interaction vector \({{{{{{{\bf{J}}}}}}}}\in {\left\{\pm 1\right\}}^{R}\), and the spins _σ__i_

form the spin vector \({{{{{{{\boldsymbol{\sigma }}}}}}}}\in {\left\{\pm 1\right\}}^{L}\). The ground-state energy for this Hamiltonian is zero, which corresponds to \({J}_{a}{\sigma

}_{{a}_{1}}{\sigma }_{{a}_{2}}{\sigma }_{{a}_{3}}=1\) for all _a_. Using the mapping \({J}_{a}={(-1)}^{{y}_{a}}\) and \({\sigma }_{i}={(-1)}^{{x}_{i}}\), we see that Eq. (3) is zero whenever

\({y}_{a}={x}_{{a}_{1}}+{x}_{{a}_{2}}+{x}_{{a}_{3}}\,{{{{{{\mathrm{mod}}}}}}}\,\,2\) for all _a_, which is Eq. (1). This lets us express the number of ground states

\({{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{GS}}}}}}}}}\) to Eq. (3) in terms of Y and _B_. Because each of the _N_Y = 2rank(_B_) vectors Y has

\({{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{GS}}}}}}}}}\) ground states (out of a possible 2_L_), we find \({{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{GS}}}}}}}}}={2}^{L-{{\mbox{rank}}}(B)}\).

The ground-state entropy is \({S}_{{{{{{{{\rm{GS}}}}}}}}}(B)\equiv \log {{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{GS}}}}}}}}}=\left[L-\,{{\mbox{rank}}}\,(B)\right]\log \!2\) (we take the

natural logarithm). QUANTUM STATE AND ENTANGLEMENT ENTROPY After applying the circuit given by _B_ (Fig. 1c) to our input state \({\left|\psi \right\rangle }_{{{{{{{{\rm{in}}}}}}}}}\), we

get the following state (see ref. 1 for more details): $${|\psi \rangle

}_{{{{{{{{\rm{out}}}}}}}}}=\frac{1}{\sqrt{{N}_{{{{{{{{\bf{y}}}}}}}}}}}\mathop{\sum}\limits_{{{{{{{{\bf{y}}}}}}}}}|{{{{{{{\bf{y}}}}}}}}

\rangle|\{{{{{{{{\bf{x}}}}}}}}\,:\,B{{{{{{{\bf{x}}}}}}}}={{{{{{{\bf{y}}}}}}}}\} \rangle \,.$$ (4) This is a superposition of solutions \(\left\{{{{{{{{\bf{x}}}}}}}}\right\}\) for each of the

_N_Y possible Y. We then measure the state of the first ∣_M_∣ parity qubits to be Yout,_M_. The resulting state is $${|\psi \rangle

}_{{{{{{{{\rm{out}}}}}}}},M}=\frac{1}{\sqrt{{N}_{{{{{{{{\bf{y}}}}}}}}}/{N}_{{{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M}}}}\mathop{\sum}\limits_{\{{{{{{{{\bf{y}}}}}}}}\,:\,{{{{{{{{\bf{y}}}}}}}}}_{|M|}={{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M}\}}|{{{{{{{\bf{y}}}}}}}}

\rangle|\{{{{{{{{\bf{x}}}}}}}}\,:\,B{{{{{{{\bf{x}}}}}}}}={{{{{{{\bf{y}}}}}}}}\}\rangle \,,$$ (5) where the first ∣_M_∣ components of Y are Y∣_M_∣ = Yout,_M_. The state is still an equal

superposition of solutions for each Y, but now there are only \({N}_{{{{{{{{\bf{y}}}}}}}}}/{N}_{{{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M}}\) terms in the sum, with

\({N}_{{{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M}}={2}^{{{{{{{{\rm{rank}}}}}}}}({B}_{M})}\). The coefficient \({\lambda

}_{{{{{{{{\bf{y}}}}}}}}}=\sqrt{{N}_{{{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M}}/{N}_{{{{{{{{\bf{y}}}}}}}}}}\) determines the entanglement entropy for such a state: $$S({|\psi \rangle

}_{{{{{{{{\rm{out}}}}}}}},M})\equiv -\mathop{\sum}\limits_{\{{{{{{{{\bf{y}}}}}}}}\,:\,{{{{{{{{\bf{y}}}}}}}}}_{|M|}={{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M}\}}{\lambda

}_{{{{{{{{\bf{y}}}}}}}}}^{2}\log ({\lambda }_{{{{{{{{\bf{y}}}}}}}}}^{2})=\left[\,{{\mbox{rank}}}(B)-{{\mbox{rank}}}\,({B}_{M})\right]\log \!2.$$ (6) The entanglement entropy coincides with

_S_GS(_B__M_)—the ground-state entropy of the classical spin model—when we choose our initial _B_ to satisfy rank(_B_) = _L_. In the limit of large system sizes, the expression for the

averaged entropy density23 is: $$\mathop{\lim }\limits_{L\to \infty }\frac{\left\langle S({|\psi \rangle }_{{{{{{\rm{out}}}}}},M})\right\rangle }{L}=\left[(1-q(\alpha ))(1-\log (1-q(\alpha

)))-\alpha (1-{q}^{3}(\alpha ))\right]\log \! 2,$$ (7) where _q_(_α_) is the spin glass order parameter after performing the ensemble average using Eq. (2). Equation (7) establishes the

exact correspondence between the entanglement entropy and the spin glass order parameter. QUANTUM HARDWARE We used the _ibm_washington_, _ibmq_brooklyn_ and _ibm_hanoi_ quantum processors24

for our experiments. We set the repetition delay to 0.00025s. We chose connected lines of qubits to take advantage of the one-dimensional structure of our circuits, while also having low

measurement readout error and CNOT error rates at the time of job submission. CIRCUIT OPTIMIZATION Because errors dominate the output in current quantum processors, we optimize our circuits

to use as few gates as possible. As the CNOT is the native two-qubit gate on the IBM Q processors, we use the CNOT count _N_CNOT as our metric (we ignore the _L_ single-qubit Hadamard gates

we always need). We build our circuits using the matrix _B__M_ since it generates the solutions we use in Eq. (2) and requires fewer gates to implement than _B_. We then decompose SWAP gates

in our circuits as $$\,{{\mbox{SWAP}}}(i,\, j)={{\mbox{CNOT}}}(i,\, j)\times {{\mbox{CNOT}}}(j,\, i)\times {{\mbox{CNOT}}}\,(i,\, j),$$ (8) where _i_ and _j_ are the qubits participating in

the gate and CNOT(_i_, _j_) means qubit _i_ controls the target qubit _j_. For the “1” gate in Fig. 1, using Eq. (8) reveals two consecutive CNOT gates with the same control and target,

which we remove because they have no overall effect. As such, a one in the matrix requires two CNOTs while a zero requires three. How many qubits and CNOT gates do we need to build circuits

such as in Fig. 1c using _B__M_? There are ∣_M_∣ = _L__α_ linear equations, so we require _N_ = _L_ + ∣_M_∣ = _L_(1 + _α_) qubits. To calculate _N_CNOT, note that each row of _B__M_ contains

_p_ ones and _L_ − _p_ zeros. There are then 2_p_ + 3(_L_ − _p_) CNOTs per row of _B__M_, where _p_ = 3 for our model. Summing the CNOT count over all rows, we find

\({N}_{{{{{{{{\rm{CNOT}}}}}}}}}({B}_{M})=|M|\left[2p+3(L-p)\right]=3L\left(L-1\right)\alpha\). By transforming _B__M_ using matrix row operations, we can reduce _N_ and _N_CNOT. We begin by

putting _B__M_ into row echelon form. Then for each row of the matrix (starting from the second), we find the index of the leading one, and add this row to the rows above it which have a

zero at that index. These row additions generate more ones in the matrix, which we prefer because they require less gates than zeros to implement. We call the resulting matrix

\({B}_{M^{\prime} }\). Note that solutions to Eq. (1) for a given parity vector remain unchanged under row operations. For example, consider the following matrix:

$${B}_{M}=\left(\begin{array}{llllll}1&0&1&0&0&1\\ 0&1&0&1&0&1\\ 0&1&1&1&0&0\\ 0&1&1&0&1&0\\

1&1&0&0&0&1\end{array}\right).$$ (9) Applying the operations described gives $${B}_{M^{\prime} }=\left(\begin{array}{llllll}1&1&1&1&1&0\\

&1&1&1&1&0\\ &&1&1&1&1\\ &&&1&1&0\\ &&&&1&0\end{array}\right),$$ (10) where the omitted entries are zeros. The

form of \({B}_{M^{\prime} }\) helps us save qubits and gates. First, the matrix has size \(|M^{\prime}|\times L\), with \(|M^{\prime} |={{{{{{{\rm{rank}}}}}}}}({B}_{M})\). The resulting

circuit requires _N_ = _L_ + rank(_B__M_) ≤ _L_ + ∣_M_∣ qubits, fewer than the circuits built from _B__M_ when rank(_B__M_) < ∣_M_∣. Second, notice that in Fig. 1c, there are gates for

each entry of the matrix. By interspersing the parity and variable qubits instead of separating them, we can avoid including gates for the zeros to the left of the leading ones in

\({B}_{M^{\prime} }\). Each gate in the primitive circuit (Fig. 1b) exchanges the positions of the qubits it acts upon. The result is that a parity qubit exchanges positions with every

variable qubit. However, only “1” gates contribute to the parity we want to measure. Therefore, once a parity qubit encounters all the “1” gates for its linear equation, we measure it in

that position. We take advantage of this by altering the primitive circuit: we start the parity qubit at the top, reverse the gate sequence, and invert the control and target of the CNOTs in

each “1” gate. Then, the locations of the leading ones in \({B}_{M{\prime} }\) provide the end positions for measuring the parity qubits. For example, the parity qubit for the first row in

Eq. (10) exchanges positions with all variable qubits before we measure it. The parity qubit for the second row exchanges positions with variable qubits 6, 5, 4, 3, and 2 before measuring,

and so on. We provide an upper bound for \({N}_{{{{{{{{\rm{CNOT}}}}}}}}}({B}_{M{\prime} })\). We count the entries in \({B}_{M{\prime} }\) to the right of (and including) the main diagonal.

We assume the leading ones are all part of the main diagonal. With this assumption, the number of entries to the right of (and including) the leading ones in a _P_ × _Q_ row echelon matrix

is: $$U(P,\, Q)=\mathop{\sum }\limits_{i=1}^{P}\left(Q-[i-1]\right)=P\left(Q-\frac{1}{2}\left(P-1\right)\right).$$ (11) In our case, _P_ = rank(_B__M_) and _Q_ = _L_. There is at least a one

per row (else the row would not be a part of \({B}_{M{\prime} }\)). Since zeros contribute more to _N_CNOT, we take the worst-case scenario where all other entries are zero. This implies

there are _P_ ones in the matrix, so there are _Z_ = _U_(_P_, _Q_) − _P_ zeros. Using these results, we get the upper bound: $${N}_{{{\mbox{CNOT}}}}({B}_{M^{\prime} })\le

3Z+2P=\frac{3}{2}{{{{{{{\rm{rank}}}}}}}}({B}_{M})\left[2L-{{{{{{{\rm{rank}}}}}}}}({B}_{M})+\frac{1}{3}\right]\le \frac{1}{2}L\left(3L+1\right),$$ (12) where we get the final inequality by

maximizing the previous expression with respect to rank(_B__M_). Finally, rather than putting a variable qubit in its initial superposition \(\left |+\right\rangle=H\left|0\right\rangle\) as

an input to the circuit, we apply the Hadamard gate _H_ only when the corresponding variable first participates in a linear equation. (For example, in Eq. (10) variable 6 is first part of

an equation in row 3.) Doing so reduces errors from trying to maintain superpositions for too long in current quantum processors. It also simplifies any SWAP gate involving a variable qubit

in the state \(\left|0\right\rangle\). If we have qubits _i_ and _j_ with the latter in the state \(\left|0\right\rangle\), Eq. (8) reduces to $$\,{{\mbox{SWAP}}}(i,\, \, j){|}_{{{\mbox{}}}j

\, {{{{{{{\rm{in}}}}}}}} \, \left|0\right\rangle {{{}}}}={{\mbox{CNOT}}}(i,\ j)\times {{\mbox{CNOT}}}\,(j,\ i).$$ (13) Our circuit optimization provides a significant savings compared to

the circuits built from _B__M_, which require _N_CNOT(_B__M_) = 3_L_(_L_ − 1)_α_ gates and \(N=L\left(1+\alpha \right)\) qubits. In practice, our largest experiments (_L_ = 24 and _α_ ≳ 1)

required an average of \({N}_{{{{{{{{\rm{CNOT}}}}}}}}}({B}_{M^{\prime} })\approx 600\) gates, which is much less than the _N_CNOT(_B__M_) ≳ 1600 gates we would need if we used the _B__M_

matrices instead. ERROR MITIGATION AND SHOT COUNT In our experiments, we only keep measurement results (shots) X and \({{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M^{\prime} }\) which

satisfy \({B}_{M^{\prime} }{{{{{{{\bf{x}}}}}}}}={{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M^{\prime} }\). This provides significant error mitigation (see Supplementary Fig. 1) as we

increase the number of qubits. For \(L=\left\{8,\,16,\,24\right\}\), we took \(\left\{10000,\,25000,\,750000\right\}\) shots per sample (see next section) to obtain our data. DATA COLLECTION

1. For the desired number of matrix samples: * (a) Generate a random matrix _B_ as described in the “Entanglement phase transition” section with _L_ columns and \(L{\alpha }_{\max }\) rows.

Each _B_ is a sample and provides data for \(\alpha \in \left(0,\, {\alpha }_{\max }\right]\). * (b) For each \(\alpha \in \left(0,\, {\alpha }_{\max }\right]\): * i. Take the submatrix

_B__M_, consisting of the first ∣_M_∣ = _L__α_ rows of _B_. * ii. Put _B__M_ into row echelon form and perform row operations as described in the “Circuit optimization” section. The result

is \({B}_{M^{\prime} }\). * iii. Build the circuit corresponding to the matrix \({B}_{M^{\prime} }\) using the techniques described in the “Circuit optimization” section. * iv. Execute the

circuit on the quantum processor a sufficient number of times. Here, sufficient means measuring several pairs \(({{{{{{{\bf{x}}}}}}}},\,

{{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M^{\prime} })\) that pass the test in the following step. We always had at least 18 pairs. * v. Test measurements \(({{{{{{{\bf{x}}}}}}}},\,

{{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M^{\prime} })\) by verifying if \({B}_{M^{\prime} }{{{{{{{\bf{x}}}}}}}}={{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M^{\prime} }\). * vi.

Save the variable and parity vectors X and \({{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M^{\prime} }\) that pass the test. CALCULATING THE ORDER PARAMETER 1. For each \(\alpha \in

\left(0,\, {\alpha }_{\max }\right]\): * (a) For each matrix _B_ from the previous section: * i. Compute \({B}_{M^{\prime} }\) as in the previous section using _B__M_ with ∣_M_∣ = _L__α_. *

ii. For each saved parity vector \({{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M^{\prime} }\) associated to \({B}_{M^{\prime} }\): * A. Fix a reference solution Z that maps solutions from

the parity vector \({{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M^{\prime} }\) to the parity vector 0. We chose our reference to be the solution to \({B}_{M^{\prime}

}{{{{{{{\bf{z}}}}}}}}={{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M^{\prime} }\) whose binary form represents the smallest integer. Note that finding a reference is efficient. * B. Map

the saved solutions X associated with \({{{{{{{{\bf{y}}}}}}}}}_{{{{{{{{\rm{out}}}}}}}},M{\prime} }\) to \({{{{{{{\bf{x}}}}}}}}^{\prime}={{{{{{{\bf{x}}}}}}}}+{{{{{{{\bf{z}}}}}}}}\). Now,

\({B}_{M^{\prime} }{{{{{{{\bf{x}}}}}}}}^{\prime}={{{{{{{\bf{0}}}}}}}}\). Remove any duplicates. Call this set \(X=\left\{{{{{{{{\bf{x}}}}}}}}^{\prime} \right\}\). * iii. Compute

\(q({B}_{M^{\prime} })\) in Eq. (2) by uniformly sampling \(\min \left(24,| X | \right)\) solutions from _X_, where we determined the number 24 yields a reasonable compromise between

accuracy and quantum runtime. * (b) Compute _q_(_α_) by averaging over \(q({B}_{M^{\prime} })\) for all \({B}_{M^{\prime} }\). Following the procedure for each _L_ produces the curves in

Fig. 3. To calculate the order parameter in step iii), we want as many solutions \({{{{{{{\bf{x}}}}}}}}^{\prime}\) as possible, but a finite number works, making the order parameter

efficient to measure. For the classical simulation of _q_ (the dashed lines in Fig. 3), we uniformly sample \(\min \left(24,\, {{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{GS}}}}}}}}}\right)\)

solutions to the equation _B__M_X = 0, where \({{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{GS}}}}}}}}}={2}^{L-{{{{{{{\rm{rank}}}}}}}}({B}_{M})}\) is the total number of solutions. We did this by

sampling random linear combinations of the basis vectors forming the null space of _B__M_. This is also efficient. We note that the small size of the sample _X_ leads to appreciable

artefacts in Fig. 3, such as a deviation of the order parameter from the expected value of zero at small _α_. Concomitantly, we notice the onset of finite-size effects at values of _L_ and

_α_ for which \({\min} \left({24},{\,} {{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{GS}}}}}}}}}\right) \approx {{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{GS}}}}}}}}}\). The dip of _q_ at small _α_

for _L_ = 8 is such an effect. We nevertheless notice that, for all _L_ and _α_, theory and experiment match well in Fig. 3, since these effects are present in both. FINITE-SIZE SCALING The

objective of finite-size scaling is to take the data in Fig. 3 and try to collapse it onto a common curve by finding suitable critical parameters. We follow the technique of ref. [25] and

our previous work1. In particular, we use the scaling form: $$q=f\left(\left(\alpha -{\alpha }_{c,\exp }\right){L}^{1/{\nu }_{\exp }}\right),$$ (14) and we minimize a cost function with the

data and its associated error as input to find the critical parameters \({\alpha }_{c,\exp }\) and \({\nu }_{\exp }\). We store our experimental data as a triple \(\left(\alpha,q(\alpha

),e(\alpha )\right)\), where _e_(_α_) is the standard error of the mean for the data point. The standard error of the mean for _n_ samples is: $$e(\alpha )=\sqrt{\mathop{\sum

}\limits_{i=1}^{n}\frac{{\left[{q}_{i}(\alpha )-\bar{q}(\alpha )\right]}^{2}}{n(n-1)}},$$ (15) where _q__i_(_α_) is the order parameter for a given matrix _B_ and _α_, and \(\bar{q}(\alpha

)\) is the mean of _q__i_(_α_) over _i_. We then transform the triple according to the scaling form: $$\left({t}_{i},\, {g}_{i},\, {e}_{i}\right)=\left(\left[\alpha -{\alpha }_{c,\, \exp

}\right]{L}^{1/{\nu }_{\exp }},\, q(\alpha ),\, e(\alpha )\right).$$ (16) We sort these triples by their _t_-values and then compute the cost function: $$C({\alpha }_{c,\, \exp },\, {\nu

}_{\exp })=\frac{1}{T-2}\mathop{\sum }\limits_{i=2}^{T-1}w\left({t}_{i},\, {g}_{i},\, {e}_{i}|{t}_{i-1},\, {g}_{i-1},\, {e}_{i-1},\, {t}_{i+1},\, {g}_{i+1},\, {e}_{i+1}\right),$$ (17) with

_T_ being the number of data points. The quantity in the summation is: $$w\left({t}_{i},\ {g}_{i},\ {e}_{i}|{t}_{i-1},\ {g}_{i-1},\ {e}_{i-1},\ {t}_{i+1},\ {g}_{i+1},\

{e}_{i+1}\right)={\left(\frac{{g}_{i}-g}{{{\Delta }}\left({g}_{i}-g\right)}\right)}^{2},$$ (18)

$$g=\frac{\left({t}_{i+1}-{t}_{i}\right){g}_{i-1}-\left({t}_{i-1}-{t}_{i}\right){g}_{i+1}}{\left({t}_{i+1}-{t}_{i-1}\right)},$$ (19) $${\left[{{\Delta

}}\left({g}_{i}-g\right)\right]}^{2}={e}_{i}^{2}+{\left(\frac{{t}_{i+1}-{t}_{i}}{{t}_{i+1}-{t}_{i-1}}\right)}^{2}{e}_{i-1}^{2}+{\left(\frac{{t}_{i-1}-{t}_{i}}{{t}_{i+1}-{t}_{i-1}}\right)}^{2}{e}_{i+1}^{2}.$$

(20) The cost function \(C({\alpha }_{c,\, \exp },\, {\nu }_{\exp })\) measures, for each index _i_, the squared deviation of the point \(\left({t}_{i},\, {g}_{i}\right)\) from the linear

interpolation _ḡ_ between the points \(\left({t}_{i-1},\, {g}_{i-1}\right)\) and \(\left({t}_{i+1},\, {g}_{i+1}\right)\) on either side of the sorted sequence. We exclude the first and last

points in the sequence since they have no neighbouring points to the left or right, respectively. The uncertainty (Eq. (20)) is a weighted sum of the squared error of the current point

\(\left({t}_{i},\, {g}_{i}\right)\) and the squared error from the linear interpolation (Eq. (19)). We skip over any three identical _t_-values in a row in Eq. (17) because of the division

by zero in Equations (19) and (20) (though this only happens for isolated values of \({\alpha }_{c,\exp }\)). When this happens, we reduce the denominator of the fraction in front of Eq.

(17) by the number of skips. We plot the cost function over a grid of values near the critical parameters from the literature (Supplementary Fig. 2). This allows us to visualize both the

minimum and the uncertainty around it. The collapse for finite-size scaling works best when there are finite-size effects, so we restricted our data for the cost function to the region

_α__c_ ± 0.5 (the black connector linking the main plot with the inset in Fig. 3). We chose our grid for the critical parameters to be \({\alpha }_{c,\exp }\in \left[0.85,\, 1.10\right]\),

with a step size of 0.001, and \({\nu }_{\exp }\in \left[1.5,4.0\right]\), with a step size of 0.01. We chose a finer step size for \({\alpha }_{c,\exp }\) because we know the critical

threshold. We used a larger step size for \({\nu }_{\exp }\) because there is less precision in the literature for _ν_. To estimate our uncertainty, we plot a contour at the level

\(\left(1+r\right){C}_{{{\mbox{min}}}}\), where _r_ is the size of the maximum deviation we allow in the minimum value. We chose _r_ = 0.25, which means we remain uncertain about the minimum

for values that are up to 25% larger. Changing _r_ will grow or shrink the contour. We note in Supplementary Fig. 2 that the cost function’s minimum resides roughly in the centre of the

contour. To quantify our uncertainty, we compute the width and height of the rectangle circumscribing the contour. Then, we take the uncertainty in \({\alpha }_{c,\exp }\) to be half the

width and the uncertainty in \({\nu }_{\exp }\) to be half the height. This gives us the following experimental values for the critical point and critical exponent: $${\alpha }_{c,\exp

}=0.95\pm 0.06,\,\,\,{\nu }_{\exp }=2.5\pm 0.5.$$ (21) DATA AVAILABILITY The error-mitigated output from the quantum processors is available26 at the following Zenodo repository:

https://doi.org/10.5281/zenodo.7120441. Source data are provided with this paper. CODE AVAILABILITY We used Qiskit27 to execute the quantum circuits on the IBM Q quantum processors. All code

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https://qiskit.org/. Download references ACKNOWLEDGEMENTS This work was supported by the Ministère de l’Économie et de l’Innovation du Québec via its contributions to its Research Chair in

Quantum Computing and the IBM Q Hub of Institut quantique at Université de Sherbrooke. The work was also supported by a Natural Sciences and Engineering Research Council of Canada Discovery

grant (S.K.), a B2X scholarship from the Fonds de recherche–Nature et technologies and a scholarship from the Natural Sciences and Engineering Research Council of Canada [funding reference

number: 456431992] (J.C.). We acknowledge Calcul Québec and Compute Canada for computing resources. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Institut quantique & Département de

physique, Université de Sherbrooke, Sherbrooke, QC, J1K 2R1, Canada Jeremy Côté & Stefanos Kourtis Authors * Jeremy Côté View author publications You can also search for this author

inPubMed Google Scholar * Stefanos Kourtis View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS J.C. conducted the experiments, performed all

simulations and data analysis, made the figures, and wrote the paper. S.K. conceived the idea for the project, provided guidance along the way, and wrote the paper. CORRESPONDING AUTHOR

Correspondence to Stefanos Kourtis. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests. PEER REVIEW PEER REVIEW INFORMATION _Nature Communications_ thanks the

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criticality on a quantum simulator. _Nat Commun_ 13, 7395 (2022). https://doi.org/10.1038/s41467-022-34982-3 Download citation * Received: 06 July 2022 * Accepted: 11 November 2022 *

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