Scrambling Transition in a Radiative Random Unitary Circuit Zack Weinstein1Shane P. Kelly2Jamir Marino2and Ehud Altman1 3 1Department of Physics University of California Berkeley California 94720 USA

2025-05-03 0 0 3.64MB 20 页 10玖币
侵权投诉
Scrambling Transition in a Radiative Random Unitary Circuit
Zack Weinstein,1, ∗Shane P. Kelly,2, ∗Jamir Marino,2and Ehud Altman1, 3
1Department of Physics, University of California, Berkeley, California 94720, USA
2Institute for Physics, Johannes Gutenberg University of Mainz, D-55099 Mainz, Germany
3Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
(Dated: December 5, 2023)
We study quantum information scrambling in a random unitary circuit that exchanges qubits
with an environment at a rate p. As a result, initially localized quantum information not only
spreads within the system, but also spills into the environment. Using the out-of-time-order corre-
lator (OTOC) to characterize scrambling, we find a nonequilibrium phase transition in the directed
percolation universality class at a critical swap rate pc: for p<pcthe ensemble-averaged OTOC
exhibits ballistic growth with a tunable light cone velocity, while for p>pcthe OTOC fails to
percolate within the system and vanishes uniformly within a finite timescale, indicating that all
local operators are rapidly swapped into the environment. To elucidate its information-theoretic
consequences, we demonstrate that the transition in operator spreading coincides with a transition
in an observer’s ability to decode the system’s initial quantum information from the swapped-out, or
“radiated,” qubits. We present a simple decoding scheme which recovers the system’s initial infor-
mation with perfect fidelity in the nonpercolating phase, and with continuously decreasing fidelity
with decreasing swap rate in the percolating phase. Depending on the initial state of the swapped-in
qubits, we further observe a corresponding entanglement transition in the coherent information from
the system into the radiated qubits.
Understanding the complexity of quantum states and
operators undergoing time evolution is a key challenge
with potential implications across fields, from condensed
matter physics, through quantum gravity, to quantum
computation. In condensed matter physics, insights on
the growth of operator complexity have inspired new
ways of computing the dynamics of thermalizing sys-
tems [1–8]. Operator growth, as measured for example
by out-of-time-order correlations (OTOCs) [9–12], is also
considered as key to relating boundary to bulk dynam-
ics in the conjectured AdS/CFT correspondence [13–19].
Furthermore, understanding complexity growth in terms
of scrambling of quantum information has revealed con-
nections between the dynamics of black holes and the
capacity of artificial quantum circuits to encode and pro-
cess quantum information [20–23].
Physical observables evolved by simple models of un-
structured unitary circuits or of thermalizing many-
body Hamiltonians are expected to scramble and grow
in complexity indefinitely, or at least to astronomical
timescales [19,24]. But these models may be too sim-
plified in some cases. For example, as information is
scrambled in a black hole some of it is lost to Hawking
radiation [25]. Similarly decoherence in quantum circuits
implies that some of the information is ultimately lost,
or shared with external degrees of freedom [8,26,27].
Can there exist sharp thresholds or phase transitions in
scrambling, or in the flow of quantum information, tuned
by the rate of such loss processes?
Recently, it has been discovered that random unitary
circuits (RUCs) interspersed with local projective mea-
surements can exhibit two distinct dynamical phases,
characterized respectively by the partial protection or
rapid destruction of initially encoded quantum informa-
tion, which are separated by a continuous phase transi-
tion at a nonzero critical measurement rate [26,28–34].
However, measurements are highly nonrepresentative of
generic errors, and moreover, such measurement-induced
phase transitions (MIPTs) typically face exponentially
large postselection barriers to experimental observation
[35–37]. It is natural to ask if a phase transition in scram-
bling and information flow can occur in a RUC without
measurements, thereby avoiding the postselection prob-
lem altogether.
In this Letter, we present a simple model of a RUC
that exhibits a phase transition from scrambling to non-
scrambling dynamics. We extend previous works [40–45]
exploring operator growth in closed-system RUCs by al-
lowing the system to exchange qubits with an environ-
ment [46–49]; as a consequence, initially localized quan-
tum information not only spreads within the system, but
also spills into the environment. Using a mapping to a
classical nonequilibrium statistical mechanics model of
population dynamics, we show that the circuit exhibits
tunable scrambling: increasing the rate of qubit swaps
reduces the OTOC light cone velocity within the system
until it vanishes at a critical swap rate. At this point the
model exhibits a phase transition in the directed perco-
lation (DP) universality class [38,50–52] to a nonscram-
bling phase. For swap rates above this threshold, all local
operators initially within the system are rapidly swapped
to the environment.
In contrast to previous works on operator growth in
open systems [8,53–60], the transition in the OTOC
described here requires that an observer has access to
the swapped-out, or “radiated,” environment qubits. To
determine the implications of the transition in opera-
tor spreading on the flow of quantum information, we
arXiv:2210.14242v2 [quant-ph] 1 Dec 2023
2
UtU∗
t
U†
tUT
t
S
1S2E
E S
ES
S
1S
2E
ES2S1
S E
SE
ES
2S
1
ρE
0
ρS
2
0
ρS
0
|0|0 0|0|
• •
Psucc =
UtU∗
tUtU∗
t
S
1S2E
E S
ES2S1
S E
S1S2E
E S
ES2S1
S E
ρE
0
ρS
2
0
ρE
0
ρS
2
0
• • • •
tr(ρE
t)2=
•
•
•
•
•
•
•
•
•
•
•
•
X
0
Yx
•
•
•
•
•
•
4
p= 0.08 p= 0.16 p= 0.206 p= 0.24
x−→
t−→
10−2
100
100101102103
t
10−6
10−5
10−4
10−3
10−2
10−1
(t)
p= 0.08
p= 0.16
p= 0.206
p= 0.24
t|p−pc|ν
(t)|p−pc|θν
a)
b)
x−→
t−→
c)
FIG. 1. (a) Circuit diagram for the model studied. An initial operator X0is Heisenberg evolved via a random unitary circuit.
In between layers of unitary gates, swaps with ancilla qubits occur with probability p. The OTOC C(x, t) is obtained from the
commutator [X0(t), Yx]. (b) Top: OTOCs for typical Clifford circuit realizations, for four swap rates p, for a system of size
N= 100. White denotes C(x, t) = 1, while black denotes C(x, t) = 0. Bottom: averaged OTOC for the same swap rates in a
system of size N= 1024, depicting the narrowing and eventual vanishing of the light cone. Colors are plotted on a log scale
for increased contrast. (c) Integrated OTOC ϱ(t) = 1
NPxC(x, t) for several swap rates pin a system of size N= 1024. ϱ(t)
exhibits linear growth and saturates at a finite value for p < pc, exhibits power-law growth with exponent θ≃0.3175 at the
critical point pc≃0.206, and rapidly decays to zero for p>pc. Inset: scaling collapse for several swap rates below pc(green)
and above pc(purple), using DP exponents θDP ≈0.3136 and ν∥,DP ≈1.734 [38,39].
consider a thought experiment in which an observer at-
tempts to recover quantum information stored in the ini-
tial state of the system from the radiated qubits alone.
Motivated by an analogy to previous studies of quantum
information scrambling in black holes [21,23,47,48], we
provide a simple algorithm by which an observer with
access to the radiated qubits can decode this quantum
information with perfect fidelity in the nonpercolating
phase of the circuit, but with an imperfect fidelity set by
the DP survival probability in the percolating phase. We
numerically demonstrate a corresponding transition in
the coherent information into the radiated qubits, which
depends nontrivially on the observer’s knowledge of the
qubits swapped into the system.
Model.—We consider a one-dimensional system of N
qubits with periodic boundary conditions undergoing
brick-wall random unitary evolution [Fig. 1(a)]. Each
two-qubit unitary gate is independently drawn from ei-
ther the Haar or Clifford ensemble. Between layers of
unitary gates, each system qubit is swapped with an en-
vironmental ancilla qubit with probability p; crucially, we
use a fresh ancilla qubit for each such interaction, and we
do not trace out the ancilla following the system-ancilla
interaction. For now, we leave the initial state ρSE
0on
the system Sand environment Eunspecified.
We study operator spreading in the RUC Utby com-
puting the out-of-time-order correlator (OTOC) [9,10,
12,41,42], defined here as
C(x, t) = 1
4trnρSE
0[X0(t), Yx]†[X0(t), Yx]o,(1)
where Yxis the Pauli-Yoperator for the xth qubit, and
X0(t) = U†
tX0Utis the Heisenberg-evolved Pauli-Xop-
erator for the zeroth qubit after tlayers of unitary gates.
References [41,42] previously studied operator spread-
ing in closed-system RUCs using the OTOC. Upon in-
troducing swap gates with an environment, a new physi-
cal feature emerges: the operator X0(t) not only spreads
within the system, but also spills into the environment.
By tuning the swap rate p, the rate of operator growth
within the system can be slowed or even halted alto-
gether.
To see this concretely, it is illuminating to consider
the evolution of the OTOC in random Clifford circuits
[61,62]; see the Supplemental Material [63] for a cor-
responding calculation for Haar-random circuits. Since
Eq. (1) is second-order in Ut⊗U∗
tand the Clifford en-
semble forms a unitary 3-design, the ensemble-averaged
OTOC C(x, t) behaves identically in the Haar and Clif-
ford circuits [64,65]. Whereas X0(t) evolves into a su-
perposition of many Pauli strings in generic circuits, in a
Clifford circuit X0(t) remains a single Pauli string at all
times, and C(x, t) = 1 whenever X0(t) has Pauli content
Xor Zat site xand vanishes otherwise. Noting that
each of the three Pauli operators appear within X0(t)
at a given site with equal probability, we can express
C(x, t) = 2
3nx(t) in terms of an occupation number nx(t)
which equals one whenever X0(t) has nontrivial (i.e., non-
identity) Pauli content on site xin a given Clifford circuit.
We interpret the evolution of the ensemble {nx(t)}as a
stochastic evolution of particles on a tilted square lattice.
Vertices of the lattice sit at the center of unitary gates,
and the occupation numbers {nx(t)}at each integer time
step define a distribution of particles along the edges
3
of the lattice. The rules for obtaining {nx(t+ 1)}from
{nx(t)}are determined by noting that a two-qubit Clif-
ford gate can evolve a nontrivial two-qubit Pauli string to
any of 15 nontrivial two-qubit Pauli strings (up to phase)
with equal probability, while the trivial Pauli string al-
ways evolves to the trivial Pauli string. Upon adding
system-ancilla swap gates, the Pauli content of each site
xhas an additional probability pof swapping onto an en-
vironment qubit: effectively, the particle hops from the
system to the environment, and the Pauli content of site
xis set to the identity. We therefore obtain the follow-
ing probabilities for the propagation of particles from an
occupied vertex [66]:
t3
5(1 −p)2,1
5(1 −p) + 3
5p(1 −p), 3
5p2+2
5p,(2)
where a dark line indicates the propagation of a particle,
while a dotted line indicates no propagation of a particle.
The case p= 0 returns the results of Refs. [41,42], which
can be interpreted as the stochastic evolution of particles
which can diffuse, spread, or coalesce, but never annihi-
late; the average behavior of the OTOC can be computed
exactly in this case, and one finds a ballistic growth of
the OTOC with a light cone front that broadens as √t.
In contrast, introducing p > 0 allows particles to an-
nihilate, and yields the phenomenology of a DP problem
[38,50,51]. The OTOC’s light cone velocity continu-
ously decreases with increasing pand vanishes at critical
swap rate pc, at which point there is a phase transition
in the DP universality class [63]. For swap rates p<pc,
X0(t) percolates throughout the system qubits with a fi-
nite probability P(t) corresponding to the survival prob-
ability of the associated DP process. On the other hand,
for p > pcthe particle distribution is rapidly driven to
the absorbing state nx(t) = 0; the entire operator X0(t) is
swapped into the environment within a finite timescale,
and C(x, t) vanishes uniformly at all times thereafter.
These qualitative features are observed in Clifford nu-
merical simulations, as demonstrated in Fig. 1(b).
Note that the stochastic rules (2) are not microscopi-
cally equivalent to standard bond DP, due to correlations
between the two edges leaving a vertex. This feature is
not expected to affect the universal behavior of the two
phases or the transition, in accordance with the DP hy-
pothesis [67,68]. To confirm that the phase transition
lies within the DP universality class, we use Clifford sim-
ulations to numerically compute the integrated OTOC
ϱ(t) = 1
NPxC(x, t) for several swap rates [Fig. 1(c)]. We
find that ϱ(t) grows linearly and saturates at a nonzero
value for p<pc, while it rapidly decays to zero for p>pc.
At pc≃0.206, ϱ(t)∼tθgrows as a power law with expo-
nent θ≃0.3175, in good quantitative agreement with the
critical exponent θDP ≈0.3136 governing the analogous
growth of particle density in bond DP [39]. In the Sup-
plemental Material [63] we present additional numerical
evidence for DP universality at the phase transition.
Decoding transition.—The growth of the OTOC can
be associated with the spreading of quantum information
[12,23,69]. However, this interpretation has important
subtleties in our model. Suppose that an observer Alice
stores a k-qubit message in the initial state of the system,
which then undergoes time evolution by the circuit Ut.
It may be tempting to associate the percolating OTOC
to a capacity for another observer, Bob, to recover Al-
ice’s message from the qubits remaining in the system.
However, closer examination shows that even when local
operators develop large support on the system, they have
substantially larger support on the qubits swapped out
to the environment. As a result, Bob can never recover
any fraction of Alice’s message at long times.
To explicate the connection between operator spread-
ing and the flow of quantum information in our model,
we instead ask whether an eavesdropper Eve, who col-
lects the qubits swapped out of the system, can recover
Alice’s message. We demonstrate here that the tran-
sition in operator spreading coincides with a transition
in the fidelity with which Eve can decode Alice’s quan-
tum information from the radiated qubits using a simple
decoding protocol, shown in Fig. 2(a) and discussed be-
low. In the discussion, we comment on an analogy be-
tween our model and that of the Hayden-Preskill thought
experiment [21], and the relation between our decoding
protocol and a similar protocol proposed for Hayden and
Preskill’s problem [23].
Alice’s state is stored initially on the first kqubits of S,
denoted S1. The remaining N−kqubits of Sare denoted
S2. After evolving SE by the circuit Ut, Eve collects the
radiated qubits Eand attempts to decode Alice’s state
without access to S. To construct the decoder, Eve first
introduces an extra set of Nqubits S′=S′
1∪S′
2initialized
in an arbitrary state, then applies the reverse unitary
circuit U†
ton S′E[Fig. 2(a)]. Deep in the nonpercolating
phase, where Alice’s state is always swapped entirely into
the environment, such a decoding protocol will perfectly
reproduce Alice’s state on S′
1.
To quantify the success of Eve’s decoding for general
p, we encode Alice’s state using a reference system A
initialized in a maximally entangled state |Φ+
AS1⟩with
S1, and compute the fidelity with the same state on AS′
1
following the decoding protocol:
F(t) = tr nΦ+
AS′
1EDΦ+
AS′
1U′†
tUtρ0U†
tU′
to,(3)
where ρ0=|Φ+
AS1⟩⟨Φ+
AS1| ⊗ ρS2S′E
0is the initial
state on ASS′E, which consists of the entangled state
|Φ+
AS1⟩⟨Φ+
AS1|on AS1and an arbitrary product state
ρS2S′E
0on the remaining qubits, and U′
tdenotes the uni-
tary circuit acting on S′E. A maximal fidelity F= 1
implies that an arbitrary initial state on S1will be ex-
actly reproduced on S′
1at the end of the protocol, while
4
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
t
x
•
•
•
•
•
•
•
•
•
•
•
•
U
t
U
†
t
σ
ν
j
t
ρ
SE
0
ρ
SE
0
1
0
S
S
2
a)
b)
FIG. 2. (a) Simple decoding protocol for recovering quantum
information encoded initially within the system. Following
random unitary evolution with swap gates on SE via the
circuit Ut[Fig. 1(a)], swapped-out or “radiated” qubits are
swapped back into a second system S′via the reverse cir-
cuit U†
t. (b) Late-time log-averaged fidelity log2F(green) for
the simple decoding protocol, and late-time average coherent
information Ic(A⟩E) for an initially maximally mixed envi-
ronment (orange) and an initially pure environment (blue),
as a function of swap rate pin a system of size N= 512.
Averages are taken over 400 Clifford circuit realizations. The
dotted red line denotes pc≃0.206, as estimated from the
OTOC.
a fidelity F= 2−2kindicates that the final state on S′
1is
uncorrelated with the initial state of S1.
For simplicity, we first assume that Alice’s message
contains k= 1 qubit. Then, it is straightforward to show
that in a given Clifford circuit, F(t) simply counts which
of the operators X0(t), Y0(t), and Z0(t) are swapped en-
tirely into the environment by time t. Through the map-
ping to the stochastic model, this occurs for each such
operator with probability 1 −P1(t), where P1(t) is the
survival probability of the associated DP process initial-
ized with a single particle at t= 0. In the Supplemental
Material [63], we show that this observation results in an
average decoding fidelity F(t) = 1 −3
4P1(t) for k= 1
encoded qubit. More generally, for arbitrary kwe obtain
the bounds
1−Pk(t)[1 −2−2k]≤ F(t)≤1−P1(t)[1 −2−2k],(4)
where Pk(t) is the survival probability of kinitial parti-
cles arranged side-by-side. Notably, since F(t) is second-
order in Ut⊗U∗
t, this result holds identically in both
the Haar and Clifford ensembles. In the nonpercolating
phase p>pceach survival probability falls to zero expo-
nentially quickly, resulting in a perfect decoding fidelity
at late times as expected. On the other hand, for small
p<pcthe survival probability is large, and the decoding
fidelity is close to that of a random final state on AS′
1.
The numerical late-time behavior of log2Ffor k=Nis
shown in Fig. 2(b).
Information transition.—While we have shown that
the decoding fidelity for the simple decoder of Fig. 2(a)
undergoes the same transition as the OTOC, we are more
generally interested in the maximal amount of quantum
information Eve can recover from the radiated qubits E
using any decoding protocol. In other words, we would
like to characterize the quantum channel capacity of the
circuit Ut, regarded as a noisy quantum channel from S1
to E[70–75]. Towards this end we consider the coherent
information Ic(A⟩E) = HE−HAE from Ato E, where
HR=−tr ρR
tlog ρR
tis the von Neumann entropy of sub-
system R. The coherent information can then be used to
lower-bound the single-shot quantum channel capacity of
the circuit [70,72,75,76]. In this section it is useful to
focus on the case k=N, although the generalization to
arbitrary kis straightforward [63].
Unlike the OTOC and the decoding fidelity given
above, the behavior of Ic(A⟩E) depends strongly on the
initial state of the qubits swapped into the system. First
suppose that the swapped-in qubits are initialized in the
maximally mixed state, ρE
0=1
2NE1, where NE∼pNt is
the total number of swaps; physically, this corresponds
to the case in which Eve has no prior knowledge of
the qubits swapped into the circuit. Then, one can
show diagrammatically [63] that the subsystem purity
tr (ρAE
t)2= 2N−NEFis proportional to the decoding
fidelity. This in turn implies that Ic(A⟩E) = N+ log2F
in individual Clifford circuit realizations. Although the
logarithm prevents a simple statistical mechanics inter-
pretation for the average coherent information in either
Clifford or Haar random circuits, we observe numerically
[Fig. 2(b)] that Ic(A⟩E) in the Clifford ensemble exhibits
the same qualitative features as N+ log2Fand under-
goes a transition at the same critical swap rate as the
fidelity.
In contrast, suppose now that the swapped-in qubits
are initialized in a definite pure state, ρE
0=|0⟩⟨0|⊗NE.
Physically, this case occurs when Eve has perfect knowl-
edge of the initial state of each swapped-in qubit. Since
the global state is now pure, we can compute Ic(A⟩E) =
−Ic(A⟩S) by tracing over the swapped-out qubits in E,
upon which the swap operation becomes equivalent to
an amplitude-damping channel [77]. The system den-
sity matrix ρS
ttherefore evolves via a strictly contractive
quantum channel with a unique fixed point, and rapidly
forgets its initial conditions and approaches a unique
steady state. As a result, we expect ρAS
t≃ρA
t⊗ρS
t
to rapidly factorize into a product state. This implies
that Ic(A⟩S) = −Nafter a finite timescale, indicating
that no information can be transmitted from Alice to
Bob through the system as expected. But for a globally
pure state, this immediately suggests Ic(A⟩E) = Nis
maximal. We confirm numerically [Fig. 2(b)] that in this
case the coherent information is indeed maximal for any
p > 0. Physically, this result implies that Eve can in
principle use her knowledge of the swapped-in qubits to
decode Alice’s information from the radiated qubits for
any p > 0 [63].
Discussion.—We have demonstrated a DP phase tran-
sition in the operator dynamics and the flow of quantum
摘要：

ScramblingTransitioninaRadiativeRandomUnitaryCircuitZackWeinstein,1,∗ShaneP.Kelly,2,∗JamirMarino,2andEhudAltman1,31DepartmentofPhysics,UniversityofCalifornia,Berkeley,California94720,USA2InstituteforPhysics,JohannesGutenbergUniversityofMainz,D-55099Mainz,Germany3MaterialsSciencesDivision,LawrenceBer...

展开>> 收起<<
Scrambling Transition in a Radiative Random Unitary Circuit Zack Weinstein1Shane P. Kelly2Jamir Marino2and Ehud Altman1 3 1Department of Physics University of California Berkeley California 94720 USA.pdf

共20页,预览4页

还剩页未读， 继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

相关推荐

更多
立即下载
分类：图书资源 价格：10玖币 属性：20 页 大小：3.64MB 格式：PDF 时间：2025-05-03

开通VIP享超值会员特权

  • 多端同步记录
  • 高速下载文档
  • 免费文档工具
  • 分享文档赚钱
  • 每日登录抽奖
  • 优质衍生服务

作者详情

相关内容

更多

热门标签

人际关系 动力学连接体 力的合成 配电装置 高考理综 全宋诗作者索引 公务员考试
/ 20
评分 收藏
分享
立即下载
客服
关注